Annie's A (Disintegrate) get his mana back if the ennemy is killed : so use it on minions for an easy farm. Moreover, using it will allows you to have the passive unlocked (the stun) so you can directly stun ennemies and kill them rapidly.
Remember you do not have many HP, so try to stay far frome your ennemies and look out for ganks.

Annie is a machine gank ! When you have your Tibbers (ulti), try to launch some disintegrate on minions and unlock the stun. When you have it, go to top/mid/bot and use Stunny Tibbers ! Stunny Tibbers is really awesome because once Tibbers arrive, he uses his fists against your ennemy (more damage) and everything around him burns (more more damage). And finally, finish your ennemy with Incinerate or Disintegrate and you have a free kill ! :)

Let's do it quickly :

Sorcer's boots for speed and penetration
Rilay for epic power and health
Will of the ancient for spell vamp and power
Rabadon to increase your power from 200 to 400
Morellos to have faster spells (Annie's A already have a cooldown of 3 seconds, so if it is reduced... EPIC)
And Void staff to increase power and penetration at the end of the game.

Isn't that just OP ?

All Runes and Masteries are turn to power/power/power/power/power.
Mana is important just at the begginning so just buy some potions !

Last chapter, for killing ennemies, just follow the guide :

Get you Shield
Stun with Ulti
Burn with Incinerate
Disintigrate with Disintigrate

If the ennemy is not dead, please use your summoner's spells !
You can use as summoners spells :
Ignite + Light jump
Ignite + Ghost
Ignite + Clarity

You can mix it to have for exemple Ghost+Clarity :)

Good game with you little child ! :D