I'm never quite sure about how to handle these things.

Let's say that a hostile Wizard casts invisibility on itself on its first turn of the combat. Do you leave its token/miniature there for the players to see? What about if it moves?

I find this a really unclear part of the game.

1. I want to melee attack the invisible wizard. Do I move to a space and declare "I attack the square here?" or "I think he's here, can I attack?" We know that the wizard can be attacked, since attacks have disadvantage.
2. What about if I want to shoot at it with a bow? Do I nominate a space? A line of fire? Or do I simply know where the wizard is, despite it being invisible?
3. What if the wizard began the fight invisible? Do I make a stealth check for it and compare to passive perception?
4. After the wizard turns invisible and moves, should he make a stealth check opposed by all enemy creatures' passive perception scores?
5. What if one character's perception is high enough to notice its breathing/footsteps, but the others fail - can that PC shout "He's exactly there!" and direct their attacks?
6. If you don't allow Invisible creatures to be detected this way, how do you deal with a flying creature casting spells from greater invisibility 

I find Invisible monsters and enemies so problematic to deal with that I just shy away from using them altogether.

Quote from Sanvael >>

I'm never quite sure about how to handle these things.

Let's say that a hostile Wizard casts invisibility on itself on its first turn of the combat. Do you leave its token/miniature there for the players to see? What about if it moves?

I find this a really unclear part of the game.

1. I want to melee attack the invisible wizard. Do I move to a space and declare "I attack the square here?" or "I think he's here, can I attack?" We know that the wizard can be attacked, since attacks have disadvantage.
2. What about if I want to shoot at it with a bow? Do I nominate a space? A line of fire? Or do I simply know where the wizard is, despite it being invisible?
3. What if the wizard began the fight invisible? Do I make a stealth check for it and compare to passive perception?
4. After the wizard turns invisible and moves, should he make a stealth check opposed by all enemy creatures' passive perception scores?
5. What if one character's perception is high enough to notice its breathing/footsteps, but the others fail - can that PC shout "He's exactly there!" and direct their attacks?
6. If you don't allow Invisible creatures to be detected this way, how do you deal with a flying creature casting spells from greater invisibility 

I find Invisible monsters and enemies so problematic to deal with that I just shy away from using them altogether.

I'm interested as I have a Roll20 game coming up where the creature starts off in a darkness spell and I'm trying to figure out how to run it on the map so I totally get where you are coming from.   (There is a current thread on Darkness spell where it breaks off into a tangent on some of your questions.)

These are my thoughts that I know will get stomped on by others but I really like this topic and want to discuss it further.

1) Yeah, I'm thinking it the pick a square where the attack will occur at there is no cannot attack here or there.

2) This was my question with the darkness spell that is a rule in DMG about shooting in a line the RAW is that you have to pick a location where the arrow would hit.  Thinking more on the line of fire kind of puts the Invisible creature in a bind because if you have four players all breaking out range weapons and shooting down a separate line on the map the odds are pretty good you'll hit the wizard.  

3) I would because it gives you a benchmark and this would determine the surprise condition if the wizard makes the first attack say range spell against the players.  Though I believe when you perform an attack the invisibility is 'turned off'.

4) As above I would as it gives you a benchmark of how sneaky the Wizard is and if the players have a sense there is more going on in the room than is expected.   

5) I would say the player knows something is not right in the room but does not necessarily know where the wizard could be.  

A question 7 could be if you are moving around the map trying to find the invisible creature how would it be handled if you bump into it as you are crossing over the same occupied square.

Quote from Sanvael >>

I'm never quite sure about how to handle these things.

Let's say that a hostile Wizard casts invisibility on itself on its first turn of the combat. Do you leave its token/miniature there for the players to see? What about if it moves?

I find this a really unclear part of the game.

1. I want to melee attack the invisible wizard. Do I move to a space and declare "I attack the square here?" or "I think he's here, can I attack?" We know that the wizard can be attacked, since attacks have disadvantage.
2. What about if I want to shoot at it with a bow? Do I nominate a space? A line of fire? Or do I simply know where the wizard is, despite it being invisible?
3. What if the wizard began the fight invisible? Do I make a stealth check for it and compare to passive perception?
4. After the wizard turns invisible and moves, should he make a stealth check opposed by all enemy creatures' passive perception scores?
5. What if one character's perception is high enough to notice its breathing/footsteps, but the others fail - can that PC shout "He's exactly there!" and direct their attacks?
6. If you don't allow Invisible creatures to be detected this way, how do you deal with a flying creature casting spells from greater invisibility 

I find Invisible monsters and enemies so problematic to deal with that I just shy away from using them altogether.

You've asked a question which delves into areas where the rules are insufficient to cover what happens and worse, a variety of people on the internet have very strong opinions about it.  I'll do my best.  I will assume you're the DM.  Whether or not you put a mini on the map depends on where you want to put the onus of mental work: a mini on the map means trusting your players not to metagame, which is hard.  This is how it works with PC invisibility, normally - the PC leaves their mini on the map and both other PCs and the DM are trusted not to metagame.  Not putting it on the map means you need to keep mental track of where it is at all times.  This gets impossible quickly if you have more than 1 invisible creature to track.

1. For both melee and ranged attacks, you nominate a space you think the target is in and try to attack.
   1. There's no RAW covering this, but you should have your player roll an attack with disadvantage even if they nominated an empty space, to avoid leaking to them if they guessed correctly or not.
   2. There is very widespread disagreement on how hard it is to guess the correct space.  I will delve into this below.
2. It's generally immaterial if the wizard began the fight invisible as opposed to becoming invisible mid-fight, with the caveat that if you don't have him move after casting, the players are likely to correctly guess where he is.
3. There is no question PCs can talk to each other on their turns, including communicating to each other where they think the wizard is. 

Ok, now let's get into the weeds on determining where the invisible creature is.  I can cite you the relevant rules, but here are the camps different DMs break down into.  At the end of the day what matters is how powerful you want invisibility to be, both in the hands of your monsters and in the hands of your players.  I will refer to the invisible creature as "the wizard", but the rules are the same for all creatures.

1. Perfect Echolocation:  If the wizard doesn't Hide, everyone always knows exactly where he is at infinite range.  If he does Hide, anyone whose Passive Perception he beats doesn't know where he is and anyone he doesn't beat does know, at infinite range.  Anyone can roll Perception using the Search action to try and determine where he is, rolling against his Stealth check result.  This is the weakest version of invisibility I know of any DMs using, and it makes invisibility nearly useless in the hands of most.  Support for it is... vocal.
2. Perfect Invisibility:  No-one knows where the wizard is without a countering ability like Faerie Fire.  This version of invisibility is extremely strong.
3. Compromise:  This requires patching the missing rules 5E hasn't got, so it makes some DMs uncomfortable.  We have standard rules for trying to notice something that can be applied to invisible creatures - the problem is we have no way to know what the DC should be, as the rules never cover it.  As you might imagine, this splits into an infinite number of sub-camps depending on how each DM has houseruled, but here are some examples; all of them are fundamentally based on the idea that you have to roll Perception or Investigation to find the wizard, and the question is how hard that is.
   1. Invisible wizards automatically apply their Passive Stealth to being heard (remember, all ability checks can be Passive, not just Perception).
   2. Custom DCs for locating a wizard based on hearing, based on arbitrary factors chosen by the DM, such as wizard weight, wizard size, floor medium (stone is louder than carpet), etc.  You can also apply these as custom modifiers to the wizard's Passive Stealth.  This is particularly popular for DMs who want creatures who are farther away to be harder to hear.

Regardless of how you rule, remember that this mechanic can start coming up at level 1 whenever the PCs want it to, since all they have to do is shut their eyes and try to fight to make you give them rules for how well they can find opponents based on non-visual senses alone.

So, here's the thing about invisibility... unless the invisible creature actually takes the Hide action, then the players are still aware of basically where the invisible creature is... at least enough to target them with attacks (at disadvantage). So in that case I would leave their mini on the board and just note them as invisible, and I would only remove the mini if the invisible creature successfully hides.

Knowing my players they'll figure out that someone is hidden/invisible in the room and then walk.  Only to let the Sorcerer cast Shatter in as many locations to get complete coverage of the room hoping to break the creatures' concentration.

I leave invisible characters on the board because everyone knows where they are.  I may or may not remove hidden characters depending on how and where they hide. 

Unless you're houseruling a buff to invisibility that keeps people from knowing where creatures are, there's no reason to remove them.  It helps if you think of it as predator invisibility instead of perfect invisibility.

So what I THINK I'm learning here is an Invisible creature's position is always detected and attacks against that location are rolled with a disadvantage.  

I'm a little fuzzy on the Hide aspect of invisibility though.  If the wizard is in the middle of a room, whose location is detectable even though he is unseen could move in one direction and take the Hide action because of the heavily obscured condition he becomes undetectable?  Wouldn't moving alone give away his final position before going into Hiding so that would negate any benefit of taking that action?   Also if successfully hidden against the party members Passive Perception would by moving again then reveal the location and we are back to square one. 

I think I'm wrong here but could not tell you why.  

The Hide action basically assumes that there is some form of cover... if there is nothing to hide behind, then the Hide Action is not available, no matter how well you roll. Turning Invisible gives a creature the option to Hide even in the middle of a wide open field. So if a creature chooses to take the Hide action while invisible in the middle of the room, the Hide action basically represents them putting additional effort into stepping lightly, avoiding disturbing dust, and otherwise taking deliberate action to keep their true location obscured beyond what the invisibility alone grants. So a Wizard with low DEX might turn invisible, then attempt to Hide and roll poorly, so they're still attempting to hide but accidentally leave too much evidence of the direction they're moving and their general location is still known.

I think one thing to keep in mind is that a creature occupies a 5 foot square... most creatures are not actually 5 feet wide, so they only physically fill a portion of the space they occupy. When a creature is Invisible but not actively Hiding, they produce enough sound or other signs of their presence for reasonably perceptive creatures to be able to identify the rough 5x5 area that they're within. They don't know exactly where the invisible creature is, but they know enough to make a reasonable guess at where they should aim their attacks.

Quote from TransmorpherDDS >>

The Hide action basically assumes that there is some form of cover... if there is nothing to hide behind, then the Hide Action is not available, no matter how well you roll. Turning Invisible gives a creature the option to Hide even in the middle of a wide open field. So if a creature chooses to take the Hide action while invisible in the middle of the room, the Hide action basically represents them putting additional effort into stepping lightly, avoiding disturbing dust, and otherwise taking deliberate action to keep their true location obscured beyond what the invisibility alone grants. So a Wizard with low DEX might turn invisible, then attempt to Hide and roll poorly, so they're still attempting to hide but accidentally leave too much evidence of the direction they're moving and their general location is still known.

I think one thing to keep in mind is that a creature occupies a 5 foot square... most creatures are not actually 5 feet wide, so they only physically fill a portion of the space they occupy. When a creature is Invisible but not actively Hiding, they produce enough sound or other signs of their presence for reasonably perceptive creatures to be able to identify the rough 5x5 area that they're within. They don't know exactly where the invisible creature is, but they know enough to make a reasonable guess at where they should aim their attacks.

 Ok, a little less fizzy.

So the opposite a Wizard with low DEX might turn invisible, then attempt to Hide and rolls high and beats everyone's Passive Perception.  Does that allow the Wizard to move, as you said cautiously, around the map unrestricted of detection until a player would take the SEARCH action, beating the Hide roll, to locate their position?  So successfully hidden wizard could leave the room undetected. 

With situations in regards to Opportunity Attacks.   In the rules, it says "You can make an opportunity attack when a hostile creature that you can see moves out of your reach. "

If the wizard is being flanked by two party members cast invisibility and then move away causing a trigger of an OA would still be valid (at Disadvantage attacks) because even though invisible the wizard presence is still known?   

Yes, exactly. Once a creature makes a successful hide roll they get the full benefits of Stealth until they do something that ends their stealth, such as attacking, casting a spell, or moving directly into a creature's line of sight without any viable cover. That last one seems like it wouldn't be a factor for an Invisible Creature, but if they're in a situation where there's no feasible means of hiding their location then they aren't able to get the full advantage of stealth despite being invisible. Something like being in deep snow or sand, or even shallow water where there's no way to hide their footsteps.

But for the most part, if a creature turns invisible and then is able to move stealthily, they can easily escape with very, very few means of tracking them.

Oh, as for the Opportunity Attack Question... being invisible makes a creature immune to Opportunity Attacks, unless some other feature overrides it (like if a character has the Blind Fighting fighting style).

Quote from TransmorpherDDS >>

Yes, exactly. Once a creature makes a successful hide roll they get the full benefits of Stealth until they do something that ends their stealth, such as attacking, casting a spell, or moving directly into a creature's line of sight without any viable cover. That last one seems like it wouldn't be a factor for an Invisible Creature, but if they're in a situation where there's no feasible means of hiding their location then they aren't able to get the full advantage of stealth despite being invisible. Something like being in deep snow or sand, or even shallow water where there's no way to hide their footsteps.

But for the most part, if a creature turns invisible and then is able to move stealthily, they can easily escape with very, very few means of tracking them.

Getting Clearer (Sorry, but this has been very helpful)

Setup: middle of a throne room a Wizard finds himself surrounded by two fighters.

Round 1

Wizard: Cast Invisibility

Fight 1: Attacks Wizard at Disadvantage-- Misses

Fight 2: Attacks Wizard at Disadvantage-- Hits

Wizard: Concentration Check-- Success

Round 2

Wizard: Moves away north 20 feet takes the Hide Action-- Rolls 14 beat both Fighters Passive Perception

Fighter 1: Takes Search Action-- Roll 12 vs 14 Fail Search

 Fighter 2: Takes Search Action--Roll 5 vs 14 Fail Search

Round 3

Wizard: Moves to the Right 10 feet moves South 20 feet-- Back in line but to the right of Fighter 1
* This is where I'm confused by your statement I bolded.  Does this mean even though I'm hidden and just to the right of the Fighter without any true cover I'm detectable?

Fighter 1: Takes Search Action-- Roll 13 vs 14 Fail Search

Fighter 2: Takes Search Action--Roll 11 vs 14 Fail Search

Round 4

Wizard: Moves South 30 feet out of the Throne Room

Fighter 1: Takes the Search Action-- Rolls 18 vs 14 Success Search sees the Wizard behind him.  Yells to Fighter 2 where the wizard is

Fighter 2: Attacks Wizard with Ranged weapon at Disadvantage-- Hits

Wizard: Concentration Check-- Fails.  Wizard is no longer invisible.

Oh, sorry, no... I just mentioned entering a creature's line of sight to explain that, even with a successful Stealth Check, environmental factors can still prevent a creature from benefitting from stealth. Invisibility allows a creature to Hide without requiring cover, but if something in the environment counteracts stealth then just being invisible doesn't make them undetectable. The most common thing that does this would be if there was something on the ground that makes their movement obvious, such as dust or snow.

In a location like a Throne Room, a Wizard who successfully hides is now fully stealthed and can't be targeted without being found using a successful search action. The Wizard can then freely move where ever they want in the room, and as long as they don't perform any actions that break stealth or invisibility, they remain hidden. After spending one round taking the hide action, a Wizard is free to Dash without breaking stealth, so they can just bolt out of the throne room if they really want to.

That said, yes, once one creature succeeds on a search to find the Wizard, they can reveal the Wizard's location to their allies for more disadvantage attacks.

I leave invisible creatures on the map for my own benefit as DM. I tag them in such a way as to remind everyone they are invisible. I also have an unpainted version of my wizard mini printed in translucent resin that I use as a stand in when he is invisible.

Yeah, Dashing can be bit hard to parse logically... it seems like something that should break stealth, but by default it doesn't. I think the best way to think of it is, when you're in front of someone and actively hiding, you have to be more careful for those first few seconds as you obscure your location. Then once you're fully hidden you don't have to move quite as precisely, since your target is looking around wildly to find where you went, instead of staring right at you and keeping an eye out for signs of movement.

Quote from TransmorpherDDS >>

Yeah, Dashing can be bit hard to parse logically... it seems like something that should break stealth, but by default it doesn't. I think the best way to think of it is, when you're in front of someone and actively hiding, you have to be more careful for those first few seconds as you obscure your location. Then once you're fully hidden you don't have to move quite as precisely, since your target is looking around wildly to find where you went, instead of staring right at you and keeping an eye out for signs of movement.

Now it's clear 

Thank you so much for taking the time to answer these questions this has helped out a lot.

Now, with what I know and circling back to the OP question.

In a battle map/VTT environment if the creature when invisible I'd keep the token on the board until it takes, and succeeds, the Hide action.  Once hidden then I'd remove the token (or in Roll20 case I'd move it to the GM Layer) and move it around the board until a successful Search is made from which I'd make the token visible again.   

I believe the answer is yes but can the wizard perform the hide action, beat everyone PP, and then move on its turn. 

Yes, any creature is capable of moving after taking the Hide Action, assuming they Hide before using all of their movement speed, and an Invisible Creature can move basically anywhere they want. 

Where are you referencing in the PHB the part where Hidden and Invisible that you are undetectable?  I believe it comes from the Hiding side bar in Chapter 7.