So what do you think will make a better champ for a newbie like me?
If you like the MMO mage style of bursting things down with flashy abilities from afar before they can kill you, the "mage" description tends toward that. Annie's free this week, and that's definitely her description, but her use in serious play requires abilities that take a while to unlock. On the other hand, she's also a viable support pick; you just go up the bottom lane with a friend and focus entirely on stunning enemy players whenever they misposition.
Kog'maw is also free right now, and can be built to do that, but a lot of his potential is in right-clicking the right target in bigger engagements, or in reading enemy movement and landing shots with a second or two of lead time.
The difficulty indicator when you click champs in game is very useful, as long as you read it as a measure of mechanical difficulty rather than odds of winning - Sona's a 1 because all of her abilities autohit within range, even though she demands a mastery of footsies to play well these days; on the flip side Nidalee's literally "land your Q that's up every 4 seconds, win the game" but is rated a 7 because she's got a couple seconds of windup/travel time on it and has an alternate melee mode with its own abilities to learn.
Now how does it work? You get the champ with starter skills or every skill but lvl 1 weak ones? Sorry for all the questions I want to begin with some kind of idea.
Each level you pick a skill and upgrade it by one point. At 6, 11, and 16 you have the opportunity to upgrade your "ult" or "R" - it's usually your most powerful ability, but is locked away until further into the game.
However, the particular ability I mentioned for Annie is one of the "summoner spells" you unlock for your account, which are just chosen pregame. As a new player you'll have things like a self-heal, a movement speed boost, etc that you can use once every few minutes; the "serious" Annie strategy relies on a short-range teleport called Flash to leap at the enemy and use her ult.
But if you're a new player, the people you're playing against probably won't have it either.
Just jumping into the tutorial is probably best. The game's full of stuff that we can take forever describing, but it'll only really make sense once you try it for yourself.