If you want to deal damage with Moonbeam on your own turn, you can use Thorn Whip to force an enemy to enter the area of effect.  This only works if the enemy is close to the area of effect, and if you can place yourself in a position where they can be dragged towards you with 10ft of movement to be pulled into the area of effect.  A mass AoE like [spell[Thunderwave[/spell]can also work to force an enemy to enter the area of effect, and it pushes them, whereas Thorn Whip pulls them.  Unlike Thorn Whip, it requires a spell slot, but it can also push multiple enemies into the Moonbeam.

If your allies have forced movement abilities, this spell also becomes a lot more powerful. 

I was also looking at that and it was actually a point of contention between me (Druid) and my DM. I'm reading it as a radius of 5ft, so a total of 10ft, but he only saw the 5ft cylinder at the top of the spell. Math would have a clear rule that its diameter is 10ft across. But as read, he's saying 5ft is all it gets. Trying hard to not be a rules lawyer here, but it really nerfs this spell for me right now.

Does anyone know why this is not a Cleric Spell?

I was also looking at that and it was actually a point of contention between me (Druid) and my DM. I'm reading it as a radius of 5ft, so a total of 10ft, but he only saw the 5ft cylinder at the top of the spell. Math would have a clear rule that its diameter is 10ft across. But as read, he's saying 5ft is all it gets. Trying hard to not be a rules lawyer here, but it really nerfs this spell for me right now.

Circular (cylinder, sphere) AoEs use radius as a characteristic. Look at Fireball. It says 20' radius and the graphics at the top shows a sphere with the number 20. You are correct in your interpretation.
From the PHB Spellcasting Areas of Effect section for cylinders:

A cylinder's point of origin is the center of a circle of a particular radius, as given in the spell description.

For spheres:

You select a sphere's point of origin, and the sphere extends outward from that point. The sphere's size is expressed as a radius in feet that extends from the point.

Question: One of my players is adamant that “a point within range” can mean the base of the beam can be in the air. So if they are attacked by flying creatures, she points to where their feet are and casts the spell so that the creatures are engulfed in the beam but the players can move around under them and attack. Is this correct?

Quote from Forevermidnight >>

Question: One of my players is adamant that “a point within range” can mean the base of the beam can be in the air. So if they are attacked by flying creatures, she points to where their feet are and casts the spell so that the creatures are engulfed in the beam but the players can move around under them and attack. Is this correct?

 No, it is not correct. While "a point within range" can be in the air, the base of a cylinder cannot be placed in the air.

The different shapes used as areas of effects for spells have specific constraints defined in the rulebook. 

You can check the definition of a cylinder area of effect here. I've quoted it here for reference and bolded the relevant part.

Cylinder

A cylinder's point of origin is the center of a circle of a particular radius, as given in the spell description. The circle must either be on the ground or at the height of the spell effect. The energy in a cylinder expands in straight lines from the point of origin to the perimeter of the circle, forming the base of the cylinder. The spell's effect then shoots up from the base or down from the top, to a distance equal to the height of the cylinder.

A cylinder's point of origin is included in the cylinder's area of effect.

The origin point has to be within the 120 ft. range of the spell, but it also has to be a valid cylinder origin point as defined by the rules, so it can't be just anywhere in the 120 ft. The point has to be on the ground (or on a stable surface/floor if there are platforms or multiple levels), or, if a point in the air is specified, it has to be 40 ft. above the ground (or a stable surface/floor).

Geometrically this could be interpreted to mean that specifying the top of the cylinder reduces the top-down 2D range of the spell below 120 ft., but you can choose to fudge that part to facilitate smoother play. On a completely flat battlefield at max range, it only makes a difference of just under 7 ft.

Why couldn't they have just said moonstone for the component part? They can say a diamond fine for a revivify spell, for erupting earth they can say a piece of obsidian very easily, but for a moonstone they have to be fancy and call it an opalescent feldspar? I don't get it 

Amazing how one misused word in a spell description can cause so much confusion. I am talking about whether the spell does damage upon casting, which is probably one of the most confusing aspects of the spell's description:

"When a creature enters the spell's area for the first time on a turn or starts its turn there..."

All they had to do is change the "a" to "its" because "a" literally means a turn, whereas "its" specifies the creature.

Rephrasing to the following would end all questions:

"When a creature enters the spell's area for the first time on its turn or starts its turn there,..."

Quote from denmanthewall >>

Amazing how one misused word in a spell description can cause so much confusion. I am talking about whether the spell does damage upon casting, which is probably one of the most confusing aspects of the spell's description:

"When a creature enters the spell's area for the first time on a turn or starts its turn there..."

All they had to do is change the "a" to "its" because "a" literally means a turn, whereas "its" specifies the creature.

Rephrasing to the following would end all questions:

"When a creature enters the spell's area for the first time on its turn or starts its turn there,..."

I've come here to clarify the point of when damage is inflicted, but I think contrary to what you're saying...

On "A" turn, it could mean that a cleric, on its turn uses COMMAND to lure an NPC target to step into the moonbeam... Therefore taking damage on the Clerics turn...

If it was, like you say,  that it had to wait until the NPC targets turn to take that Damage, then it would be a waste of a turn for the Cleric in that round.

I'm pretty new to all this, so take my interpretation with a pinch of salt, But I see moonbeam as one of those companion effect spells that encourages teams into cooperative play.

Extending the example, Cleric forces target to step into the moonbeam, target takes damage, continues approaching cleric and exits moonbeam... wizard then uses gust to force Target back into Moonbeam, thus taking further radiant damage... 

Round ends, target NPC starts in moonbeam, but as it's not it's first time in there no damage, it exits, tries moving away. Does whatever. Druid continues to concentrate giving the Cleric/Wizard the opportunity for their 1,2 (Command/gust) combo.

Maybe, I dunno. There's probably flaws in that strategy, but in principle that's how I see it working. 

If the caster loses sight of the Moonbeam, does she continue to be able to move the Moonbeam as an action?

For example: A druid casts Moonbeam and sends it into a room. Her foe closes the door to the room and bars it. On her turn, the druid chooses to maintain concentration on the Moonbeam. Can she move the Moonbeam to a specific spot in the room or would the movement be randomly determined?

Does a monster like a couatl count as a shapechanger (due to its 'Change Shape' ability), or are only monsters with the 'shapechanger' tag affected?

A shapechanger makes its saving throw with disadvantage

What constitutes a shapechanger? Is any druid player a shapechanger? What about a wizard with the polymorph spell? 

official sage advice: 

https://www.sageadvice.eu/2016/05/25/do-wild-shaped-druids-or-polymorphed-creatures-count-as-shapechangers-for-moonbeam/

personal rule on my table:
If a spell or ability said that the user reverts to his/her natural form when they are reduced to 0HP then i would qualify that as Shapechanging. 

Quote from Ozmer82 >>

Question: Is there an official answer on whether or not Moonbeam would effect the shapechanger ability of the Changeling player race?

As a DM, I would rule that it does, based on the fact that the changeling NPC has the shapechanger tag.

Quote from Bhuraelea >>

personal rule on my table:
If a spell or ability said that the user reverts to his/her natural form when they are reduced to 0HP then i would qualify that as Shapechanging. 

That makes sense. I wouldn't limit it to those kinds of abilities, though. Changeling doesn't say that, for instance (they revert on death, not on 0HP).

Edit: I would also restrict it so that it has to be an ability, and not include spells. Someone who had polymorph cast on them might be "shapechanged," but that wouldn't make them a "shapechanger," they're just a normal person who had a spell cast on them. A druid, on the other hand, would be a shapechanger according to this definition. Although maybe they wouldn't have to have disadvantage on their saving throw unless they're in wild shape, because it seems unfair to have a spell that specifically targets one class more than any other.

If the caster loses sight of the Moonbeam, does she continue to be able to move the Moonbeam as an action?

This is a good question and I'd handle it two different ways.

Fast play - once you lose sight of it, the spell is broken.

Realistic play - If you cannot see it, you cannot move it accurately. You can keep it active by concentrating obviously, but if you try to move it sight unseen, you need to make some kind of ability check, like intelligence DC 15 (scaled up or down due to familiarity of the space in question), to succeed. If you fail the check, you lose control and roll a d8 to set the direction and a d12 to see how far it travels (in 5 ft increments). If the cylinder moves into a 5ft thick wall or another impenetrable obstacle, then the spell is broken. You could also let the moonbeam stop at walls so it bounces around the room like a pinball. That could make for some fun gameplay as if the room is full of creatures and they shut the door thinking all is good, then the moonbeam starts moving around randomly like a spotlight, leaving the creatures in a trap of their own making!

I have a question, that I could not find an answer to. When it says “a creature” does it mean the caster will also take damage if in the beam?

and if I got it right - the caster will take that damage only once if staying within the beam, right?

trying to think my way out of an upcoming fight here, and thought I could use moonbeam as a shield of sorts for minor damage on a first round…

The caster definitely counts as a creature, and would follow the normal rules for when they take damage. 

I really love this spell on my Circle of the Stars Druid. 

Basically, I move it 1 Square behind a target, and then use my telecinetic shove ability to push it into the area so it gets damaged on my turn, and if it resists my shove, tough luck but my teammates can still push it in^^

do you roll for it?