Annie is great for new players, she teaches you a lot just through her gameplay (last hitting, ability combos, ect.)
Yeah, I was actually using her just a few minutes ago. I don't know alot of the technical terms about combos and stuff yet, but I do especially like how she can summon Tibbers.
Every fourth/fifth spellcast with Annie stuns everything it hits (don't remember the exact number. Read her passive). My personal favorite trick to use on people is to stay at 3 casts, then bring up my shield for the last spell and instantly summon Tibbers for the stun :)
Also note that you can get (Riot) Tristana for free by a certain link that likes LoL on Facebook (Google it) and (Unchained) Alistair for subscribing on YouTube.
Tristana's midgame is terrible, but her E harasses well early and her scaling bonus attack range on her passive turns her lategame into a high damage point-and-click adventure. Plus, Rocket Jump can save your life. Though it's pretty clumsy to use as an escape, and it's worth noting that it's actually a dash move that ignores units and walls and not a jump, so you still get hit.
"Luna, I'm afraid he's right. You can't go around calling people that, even if in this case it appears to be the truth." - Sena
First Champ I played with was Master Yi. Really strong and easy champ to play. My cousin and I started playing at the same time and before we knew anything about the game we would go Master Yi / Teemo Top or Bottom. Had a lot of fun!