I'm in the middle of DMing my first campaign ever - this is like the second time I've ever played this game. I think I'm doing good so far - but what I'm struggling with is when is it acceptable to go out of my way to give interesting loot to the player characters? Magic items? Fancy things? Shiny itty-bitty doodads? Is there a risk to giving out +1 variants of weapons too early? My player characters are currently at level 4.

So, in standard D&D, I don't give out +1 until level 5 - 9, +2 until levels 10 to 13, and +3 only above that. 

That goes for all the things that have +1, +2, +3, etc. I will note that this makes me stingy (again, in standard D&D).

I will give out the occasional other kinds of magic items according to general level (so I am not going to give a Robe of the Archmagi to a 3rd level character).

But wands, rings, potions, and the like -- the "common" stuff I start giving our around 4, moving them up in power ever four levels.

Lastly, I also give out items that they will need "down the road" if I know the overall adventure path. For me, that usual means it won't be useful in this adventure, but it will be in the next.

Now, I say "standard D&D" because we don't always play that way, lol. We have a lot of house rules and such, and so normally in our games there are more magical items and such -- but we all started playing in 1st Edition (AD&D), and we like our magic items, and so we set up rules so we could have more of them.

all of that said -- you are brand new. If you want to try it out give them something with charges -- wands, rings, or be creative. THe reason I say give them a charge based item is that once they run out of charges, it can die or stop working if you want it to do so. THis gives you the flexibility in case the item makes it too easy.

Now, there are a lot of folks here who play straight 5th edition all the time, and they can probably give you better guidance. But I want t point out that the point of it is still to have fun, and if you have fun giving out magic items go for it -- the players are not the Only Ones who should be having fun.

Loot is a very personal decision.  How much gold is just right and how much magic is just right?  

I play similar to AEDorsay.  Magic is scarce, but not non-existent in our world and treasure is also uncommon.  A magical weapon of any type (even +1) is a big deal.  If you like this idea, consider giving out consumable magical items, missiles, potions, disappearing dust, etc instead of a suit of magical armor or a weapon.  One of my favorite recent magical items I made up was stout ale, that gave a bonus to attack and damage; they only found three bottles of it but it saved their bacon in a fight. I also try to give magical items that will help with the next adventure (e.g. ring that causes a light wind to blow before they take to the open ocean on a sailing ship).  

For us, parties level 1-4 get a few hundred gold per person per adventure (if everything goes well) while levels 5-10 may get 1-2K.  

Non traditional loot should be considered as well.  My party got a famous whisky recipe (Fire Giant Fire Whisky) from a brewery that had been sacked by monsters.  They sold it for 2K to a bartender along with some starter yeast for the same brewery's ale.  How about a recipe for a potion or a document granting reprise from taxes?  A map to a trade port?

My advice would be to not open the floodgates early.  It is very hard to DM for a party where everyone has a +5 flaming sword of unlimited hit points.  You can always increase your benevolence in the future if you feel like you are being too stingy.  

It's... Interesting. I play and DM. 

5e is designed so you don't really need any magical items. (My DM is INCREDIBLY stingy with even basics like studded leather for the rogues... It's a contrast...)

The party that I'm DMing for, as soon as they got loot wanted the most obscene magical items they could think up, which I begrudgingly allowed some plus one stuff.

The thing is, an extra 1 or 2 to hit is only slightly upping your chance to hit, which can be counter by just raising the AC of things 1 or 2. it's also not nearly as bad as some of the bardic inspo's and guidances and other spells that buff to hit and so on.... Hell anything that gives advantage is broken beyond belief by comparison to just a measly 1 extra point to hit. (which is usually weapon proficiency anyhow)

As for damage output, it's 1 or 2 points. It's hardly the same as the rogue's natural sneak attack dice, or the extra attacks or the stuff ki points allow or the rages......

Because 5e is designed so magic crap doesn't matter. (within reason anyhow). 

Your every 4 level ASI's are going to do more to imbalance things.

I don't like magical weapons, they aren't needed, but let the players dream. 

I'm not really sure how far I diverge from what's "intended," but I generally give out one permanent magic item per adventure, with an adventure being a self-contained quest ("clear out the kobold lair," or "retrieve the lost Scepter of Akroman"). This averages out to about 1.5 - 2 magic items given per level, but we generally start at level 3. So a level 4 party would have maybe 2 items so far.

I have given out +1's as early as level 3 and it's fine. The bonus to hit isn't any worse than the variation you might see from rolling stats, and I'll give these out early to players who might be lagging a bit behind the others to help even things out.