Hi everyone! Since I'm a little new to League, I need some tips on how to use Teemo and Annie a lot better. Thanks!

I'm horrible with Annie....Teemo, I'm alright with.
The best tips is just to watch people play the game. Watch pros play and you'll pick up on it.
Its better to learn the game in general first and you'll get a better idea on how to play a champion.
Twitch is a pretty good site to find videos.
Look for League sites for guides, such as Lolking. Guides can only say so much, but it'll give you a general idea of a champ's weaknesses and match-ups. Once you get a general idea of how a champion works and what items to build, you should make an item slot for that champ for the specific role (mage, marksman, support, etc.). Champions can be played many ways (such as Nidalee, she can go top, mid, or bottom), so you might want to test the champion out in many different situations.
You should also really read up on the champion's abilities. You can get a general idea of how the champion plays. By reading up Teemo's abilities, you might think of him as a champion that is mostly single magic damage that can run away easily. He can easily strike the opponent without worrying about being damaged due to his blind and his ult slow. For Annie, it might be slightly harder to figure out how she works. Her passive stuns the opponent on her next spell. Most people use the passive with her ult so Tibbers can freely do damage on the opponent when he/she can't move. She's a mage that can be a bit flexible (a shield, single damage, and area of effect damage). Spamming her spells would bring her passive up more frequently.

If you want to learn Annie better, take your W first and spam it so you can stun,I usually just farm and if they try to go all in I stun them with tibbers and burst them the hell down :D

Teemo: spam ultimate. Annie: command tibbers to rape everyone.