Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
Annie is has an amazing burst and in this build we go for AP Crit for maximum damage. Dorans Ring and HP Pot to start off because Doran's ring has good balance to start off with, move to a kage's lucky pick to get extra gold / a bit of ap which will help us get sorcerer's boots and mejai faster. Then go for a scepter because we need the health because of us being squishy. After that you get the Deathcap (one of the items replacing Zhonya's Ring) You get 2 of these because they are pretty insane ap wise. By now your doran's ring should be sold and your Kage's lucky pick sold also. Now finish off with whatever you want, can be ap or Malady, I went with it because its good to deal with magic resist a bit better.
To go into someone, make sure you time your passive, Your passive makes it so every 4 spells cast, the next spell will stun. Now farm creeps with your aoe (w) and your single target (q). Once you get 2 or 3, you can engage a target, open up with (Q) because of its short cooldown, then hit them with a (W) to do massive damage, pound them with Tibbers who does insane damage when he lands, and more thereafter, then finish with another (Q). This should usually be enough to kill.
For my summoner spells I pick Exhaust Ignite because I love to get First blood. It helps pump out Mejai and Sorcerer's boots a lot faster which means you can get stacks of it quicker. For exhaust after your stun wears off you can slow them, and ignite gives you AP and does slow painful damage. Other spells you could consider are Flash, Ghost, or Clarity... For obvious reasons.
This is my first build and although its probably hard to understand, thanks for reading it, please leave constructive criticism below... <3 hope I helped you with annie!