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This is my first time playing a warlock and I'm still fairly new to tabletop games, so I could use guidance. I know that eldritch blast does 1d10 force damage, and by 17th level, it has four beams when cast.

The PHB description for the Agonizing Blast eldritch invocation (PHB, p. 110) says:

When you cast eldritch blast, add your Charisma modifier to the damage it deals on a hit.

I've looked around and many different areas say add ability score, some people say add modifier, so its all different things.

Let's say my Charisma score is 15 (a modifier of +2). When I roll to do damage for eldritch blast, am I adding 1d10 + 15 (i.e. my Charisma score), or 1d10 + 2 (my Charisma modifier)?

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Add the modifier (+2)

As you say, agonizing blast allows one to "add your Charisma modifier to the damage [*eldritch blast] deals on a hit." (PHB p.110, emphasis mine)

Your Charisma modifier is +2, as you can see from PHB p.13. (Or many of us end up memorizing the mathematical rule: "subtract ten, divide by two, round down" to get from score to modifier.)

As a side note: almost everything in the game is going to reference your ability modifier rather than your ability score. Jump distance and movement speed in the Astral Plane are the two exceptions I can think of off the top of my head. Multiclass prerequisites and push/pull/drag capacity for those who track weight, armor prerequisites... a few other things? Point being, they're few and far between.

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    \$\begingroup\$ the push/pull/carry weight limit uses full STR score as the base for its formula too \$\endgroup\$
    – MissMisinformation
    Commented Jul 10, 2019 at 18:10
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    \$\begingroup\$ Yeah, but so few people use encumbrance at all (much to my personal dismay) that I run across it even less than Astral speed! I'm rummaging around right now for something to link: I'd swear I've written something around here that enumerates all the times a score is used, rather than the modifier. Hmm... this, about +1-to-stat magic item, might be it. \$\endgroup\$
    – nitsua60
    Commented Jul 10, 2019 at 18:13
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    \$\begingroup\$ @BradFord We often get lazy and say "add your ability score" when we really mean "add your ability score modifier", simply because it's so vanishingly rare to actually use the score rather than the modifier. When we talk about "add your strength" or "plus charisma", you can pretty much assume it's the modifier. \$\endgroup\$
    – Darth Pseudonym
    Commented Jul 10, 2019 at 20:29
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Agonizing Blast says:

When you cast eldritch blast, add your Charisma modifier to the damage it deals on a hit.

So, in your example you would add your Charisma modifier (+2) to your eldritch blast.

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Always the modifier

"When you cast eldritch blast, add your Charisma modifier to the damage it deals on a hit"

You always use the stat modifier, never the score itself. That's true of almost every case where there is math to be performed in D&D5e.

Therefore, an eldritch blast would deal 1d10+2 force damage.

Thus at 17th level, you would deal up to 4x(1d10+2 force damage), with 4 different attack rolls.

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You add your modifier

The Agonizing Blast Invocation states (emphasis mine):

When you cast eldritch blast, add your Charisma modifier to the damage it deals on a hit.

That's all there is to it, you add your modifier. The Charisma score is never mentioned in the invocation so it doesn't really matter. In your case, a Charisma score of 15 corresponds to a modifier of +2 so you would roll 1d10+2.

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