If you are there it's probably because you want to be good with Annie . Well you will not be good after reading all this. Simply because I'm not giving you skills. I will be giving you a lot of tips on how I usually play Annie . So let's begin.

Pros

High damages
AoE stun
Easy farming
You have a bear!

Cons

Easily focused
No escape spell (solved with a Flash)
Shield not really that useful

Skills and combos

Skills

Pyromania it's her passive and her most useful "ability" because using Pyromania with Summon: Tibbers or Incinerate will cause an AoE stun wich can be very useful during team fight.
To use this passive you'll only have to do 4 spells then the next one will be the stun (I will explain this on the next part).

Disintegrate is her farming ability. When you'll kill a minion/champion the mana cost will be refund so you can spam this spell as much as you want but make sure to make the final hit with this ability.

Incinerate is one of her AoE stun possibility (I will go further with this on next part). It is one of her biggest damage dealer spells so make sure to have it ready on your combo.

Molten Shield is one of the spells that is the most useless on Annie because it's starting to be useful only at late-game. Anyway it's still a good way to get your Pyromania ready with it 20 mana cost.

Summon: Tibbers is THE spell. This one is able to do a AoE stun and giving a lot of damages not only on cast but after time too. If you use this spell at the wrong time you are giving a "GG" to the other team so be careful with that.

Combos

The following combo is what you have to do on 1v1 to get a max damages output.
So let's imagine that you are on mid.
If you are not level 6, except if you have your jungler ready to jump in your chance to kill is very very low... but here is your combos before level 6:
1- Let's prepare our stun so first thing do 3 (NOT 4) Q on minions (last hit would be a good idea ;) ). Now you might be "why not doing 4 and having our stun ready?"
The answer is because if you have your Pyromania ready the other will be playing more on his defensive and we don't want that.
So now that you have 3 "stack" of Pyromania we can do this easy combo:

Beginner

Advance a little bit near of your opponent and Simply by doing Molten Shield, Disintegrate now approach a bit more to be in range of your Incinerate then send him a Ignite with that you are supposed to be killing him if he don't have much Magic Resist.

Advanced player

Get in range of your opponent and send your Disintegrate and just before your Q hit launch Molten Shield so your stun will be ready before your Q hits. As your Q is on his way move in the direction of your opponent and send him Incinerate then Ignite if you was fast enough you would have enough time to send an other Disintegrate to make sure you have the kill.

This is the part where you need to understand the most because this is where you are the person killing or not.

With Annie it's really easy to be a nuker and very easy to be a feeder. If you don't want to be the feeder you will need to get skills with her but if you want to be the one getting feed you'll need the right items (and a bit of skills ^^).

Early game

As soon as you spawn you have 1 choice to make.
1- Boots and 3X Health Potion
Take this if you are against some one for who is easy to put some damages.
2- Doran's Ring
Take this if you are against some one that will have some difficulties early game to put you damages as: Veigar or Ashe or Ezreal or Karthus or Twisted Fate (if you are able to dodge his card).

Mid Game

This is the part where most of your kills will come from so making sure to have the best items is the best thing you can do.

If you are doing really good in your lane (Cs and kills) You must go for a Mejai's Soulstealer if you don't already have it. Then taking a Giant's Belt for your Rylai's Crystal Scepter then taking a Rabadon's Deathcap is a must.

Late Game

This part is where you want to counter the enemies defensive items or where you want to get some.

First of all let's finish our Rylai's Crystal Scepter then all items is depending on what items enemies have.

If they build some MR:
Void Staff is a must.

If they don't and you are still not getting focused (you are doing something wrong haha)
Why not going on with some more AP.
Items such as Lich Bane or Zhonya's Hourglass and Abyssal Mask would be great items to have.

If you are getting focused (good job ;) )
I would recommend items such as: Guardian Angel and Banshee's Veil

Now your build is done ?
Then go buy all the elixir priority to Elixir of Brilliance

Good Spells

Ghost Ignite Flash Exhaust Teleport

Those are the spells that will prevent your lane to be too pushed ( Teleport)
that will help you evade ( Ghost, Flash, Exhaust)
or that will help your combo ( Ignite, Exhaust, Ghost)

Last choice

Surge Clarity Heal

Make sure to not take these spells except for heal they will never help you.

DON'T TAKE THESE!

Smite Clairvoyance

Leave Smite for your jungle and Clairvoyance to your support...