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.
Please bear with me. This is a basic guide from a relatively new player. It is a collaboration of what I have learned from other guides and what works for me in-game. I will try to keep this updated as patches come out and I learn the ins and outs of the game a little better.
Annie's runes are somewhat flexible. By "somewhat", I mean that Greater Quintessence of Ability Power and Greater Mark of Magic Penetration are the only real required ones. I like having Greater Seal of Ability Power for some added AP and Greater Glyph of Scaling Cooldown Reduction, so that I can get Tibbers out again a little bit sooner. You're better off using AP or cooldown reduction runes for your Seals and Glyphs, because Annie doesn't have huge mana issues thanks to the mana refund from Disintegrate, but if you notice that you're having mana issues, then feel free to pop on Greater Seal of Scaling Mana Regeneration.
The Utility tree is the best for Annie. She already hits ridiculously hard, so you don't really need to go too far into the Offense tree. All of the masteries chosen are pretty self-explanatory.
Annie loves ability power. In the beginning of the game, you can either start out with Doran's Ring or get Boots of Speed with a few Health Potion. If you're solo, then I'd suggest the boots and potions, otherwise it's personal preference. You want to get your Mejai's Soulstealer as soon as possible, because you really need to generate your stacks quickly. The faster you hit 20 stacks, the faster you get that crazy burst from AP and the quicker you can pull out Tibbers! Everything else is pretty self-explanatory. Annie relies heavily on AP, so you want a whole bunch. Get Void Staff before Zhonya's Hourglass if the other team is gearing for magic resist. If the other team has a Karthus or someone else with a stupid ability that will kill you out of the blue, then get Zhonya's Hourglass for the invulnerability active.
Annie is a super-nuke, so you want to be able to push out your super-nuke at all times. Tibbers is a real swell fella, so you'll want to put points into Summon: Tibbers every chance you get. Disintegrate is good for level 1, because you can get last-hit kills with it and have your mana refunded, while building up 5 stacks for your stun. Incinerate is your primary source of damage and is a conal AoE, so you'll want that maxxed out before your armor or single-target spell. At level 7, be sure to toss a point into Molten Shield. This will give you a small defensive CD, help a little bit with your jungling in mid-game, and gives you a charge for your stun at a very low mana cost.
Utilizing Your Burst
I've found that the most effective way for Annie to burst a target down is to drop Tibbers on their head with your 5 point stun, Disintegrate them, Incinerate them, and if they're not dead by then, you can Disintegrate again and/or Ignite.
Annie and her friend Tibbers like to kill stuff, so gets spells to help her do that! Ignite and Flash are definitely the way to go. Ignite will help with kills (especially early game) and Flash can either get you in for that quick stun, catch you up if the enemy is running away with practically no health left, or save your behind if you need to hop through a wall. Exhaust isn't too bad if you need to keep someone close or need to prevent someone else from dominating you. Ghost is an alright getaway, but pales in comparison to Flash. Clarity is good if you don't know how to use your mana properly. Everything else is pretty useless, because you shouldn't be in a situation to use them.
Annie is a nuke and must be played as such. You need to gear, rune, and use your masteries to make her nuke lethal 100% of the time.