Hi
Im not pro but i have over 300 games whit annie so i think i know what im saying.
Someting you sould know is im my self wery agressive player and my build is build for that purpose it will not work for everyone but give it a try.

Leave comments and i try answer them if you disagree for someting here.

You might notice i do not use standart layout there i have 21/9/0 and reason for that is wery simple. Utility dont give that much for annie.

why not utility?
- mana regen: annie dont realy need it, in most machups annie dont even need blue puff to survive in lane.
- summoner cooldown: this is dam good one in utility only reason i can think to get utility over defence.
- puff duration: ok but i think defence is better.

What i use:
durability, weteran scars for masteries and runes i take healt/lvl.
totally it give +300hp and few point mr or armor
and everyting other is more ap like every midlane mage...

MR runes are good if you get hard hitting opponent like leblanc. but i rarely use them i only get early lvl to fire shield and build abyssal spectre to get MR.

Starting items:
Dorans ring or boots both are viable start but i have noticed after 50+ games of testing my performance whit boots on lane is way higer.
Because annie is slow champion it makes you even or faster giving you beter change to haras enemy in early game.
1 item
Sorseres boots

2 item
Deathfire or Zonya:
if you are surviving good definetly Deathfire it give you best burst in earlygame
if you get beaten or have hard times Zonya is your ansver.

3 item
if you alredy have zonya get deathfire and way around...
OR if you have Zonya and still feel like you get beeten get Rylais spectre for slow and exstra hp.

4 item
Rabaddon this poit it start to be wort to buy meny players buy it in start but do not fall for it.

5 item
Zonya, deathfire, rylay you are still missing one of them

6 item
Void staff. get it faster if enemy stak lots of mr you can take it as item 3 or 4 and skip zonya or deatfire alltogether.

Dorans Rings:
i normaly get pack to town when i have 1k money to buy 2x dorans 2x wards. it give best setup for your lvl 6 what you are about to hit by that time. and dorans rings will pay them self back try it if you dont believe. Manaregen they give you allows you so much higer greep score you even have mana to take fraits and wolfs time to time.

ITEM YOU DO NOT GET!
ROD OF AGES!! Annie is stron early to mid game and fall litle bit in end game. So dont buy item what is aimed for end game! and this item cost 3k and it gives only starts for 2k item everyone ho buy it give enemy 3 kill wort of handicap.

Disindegrate or incinerate its more guestin of your play style.
Im wery agressive player on lane and i take disindegrate first. best for killing champs
Meany palyers take incinerate first and roam other lanes its good too.

Pyromania: this is what makes you good or bad annie you need to learn to combo it back in teamfight. you need to know when you get it and where to use it.
Disindegrate: best one target damage : best for killing single champion
Incinerate : Best for clearing minion waves and giving you time to roam other lanes.

Molten shield: it gives you 20+10/lvl mr and armor! insane! full lvl it gives 60. normaly you have 150Mr 150Armor whitout realy buying any items? its some times beter what your tank have and still somene try say you are sguisy?

Summon Tibbers:
Not meny ap carry can iniate fight like annie. Dont forget to control it after all it burns and hit 100+ so its wery inportant to learn to control it.
+ it can stand way of skill shot.
+ it can guard and stay as ward if you whant to dive.
+ IT CAN TANK TOWER FOR YOU IF YOU DIVE.
+ it can out zone enemy laner and make lots of damage if hes fool and fight him.
+ tibbers burn prok rylais slow so if hes around every enemy is perma slower if they try follow you.

Note: Turret allways target first tibbers after it annie.
but if tibbers make damage to enemy champ tower change to defend and attack annie not tibbers! and dont forget tibbers burn aura get easily turret to change targets.
control it well and it tank tower for you when you dive.

and for combos:

Normal haras:
stun ready hit enemy whit Q and follow it by W and AA anf fall back.
+ lots of damage and enemy cannot retaliate when you have stun.

Counter haras:
every time enemy try hit you whit skills you whant him too whit yours when hes skill are in cooldown. and if you keep your stunn redy alltimes you can go for normal haras when you allways win.

AA haras:
hit whit AA every time when he try last hit creep wrok wery well vs melee champs ( kassadin, lee sin, talon...) work best in early lvl 1-5.

Kill:
haras until you see openig for kill.
stun whit tibbers use flas to get in range if needed follow whit deatfire grasp and Q + W + ignite.
And start runnig way where most likely enemy will escape and follow up whit anothe Q and AA.

Flash & Ignite there is no room for anyting other =/.

Flash: only repositionig skill you can have i cannot count how meny kill it have given me.

Ignite: 95% of times game have balanced so you cannot kill enemy whit one burst ignite give you that litle exstra boot to finis it.
and you might have noticer litle theme in annies build so how you can skipp this?

annies strong point is taking one champ out of game instantly so take you time and choose wisely. after all that is you most inportant job and you have only one shot in team fight.

So try find ap or ad if possible but like we all know as ap caster whitout any dash / escape tool its next to impossible if enemy dont fail.

so next best target is enemy assassin or bruiser normaly it is not wery hard to find him because he will try to find you =P. here you only need to know diference between bruiser and tank, dont waste yous skills on tank if possible.

After initial burst you dont do so much damage so your job is to take down any target you can. when you have skills redy take someone out its beter take one player out that make damage to another.

ps. keep tibbers in midle of fight dont underestimate burn damage it make.

+ Best area burst in game ( this is why you are reading this guide?)
+ not weakest ap carry (everyone say annie is sguisy champ but i dont agree)
+ game chancer passive ( you have lvl 1 stun that everyone get on lvl6 like amumu.)
+ personal bodyguard tibbers ( tibbers learn to use it)
+ Mana effective champ (annie is wery good choise if your jungle need blue puff a lot)

- no escape and get focused fast (thats why you have zonya)
- hard to position right in fight (never go win or lose fight whitout flash)
- not so good dps (You have huge burst but after it your dps drops wery badly)