This build for annie will have you bursting out a combo that takes down carries instantly, and even burns down tanks due to the huge magic penetration. The idea isn't survivability. You want to go into a fight with a stun up, and to burn down as many enemies as possible while kiting.

Keep your stun charged

As soon as the battle begins, you put one point in incinerate and start charging it up. This is something you need to get used to playing annie, as you roam the battlefield you should constantly be casting incinerate and molten shield. So that when you enter a fight you will always have the advantage with your stun.

Annie's combo:

Gear the combo to stun as many people as possible given how many stacks of pyromania you have. It is always best to begin the combo with tibbers, as he can be dealing damage while you are blasting off the rest of your combo. So generally R>W>Q But it varies greatly with situation.

Top or bottom?

You will be a key asset to taking down top if you can get a good incinerate out, and stun a few of them. But you are often far too squishy to survive the encounter. So the other option is taking bot. This is best done by picking the promote summoner spell. As soon as the big minion is about to die, cast promote and walah your lane will be pushed. Other than this, play bot lane similarly to a summoner's rift game but make sure to focus much more on harass than last hits. The great thing about annie bot lane, is she can burn down any squishy incredibly easily and move on to take the oppositions bottom point.

-> -> -> -> -> ->

CORE:

The evil tome gives much needed CD reduction, to bring that Tibbers CD down. It gives plenty of AP and enough mana regenration that you will not have to worry about mana the rest of the game.

The rest:

You can buy and in either order. Getting void staff first is a must if they have tanky team (2 or more tanks). Often which one you buy first is determined by how much gold when you recall each time. After getting morello's evil tome here is what you should do with certain gold amounts.
<1600 buy , otherwise buy
If you bought a blasting wand, do a similar thing on next recall
1400<gold<1600 buy , otherwise buy .
After that, just continue buying pure AP items, you can invest in a crystal sceptre if the game is going for a long time.