Annie is one of the best heroes. Her skills are very easy to learn/use, Her pet can block skill shots , does aoe damage, and with her special (4 STACKS!) can stun a whole team! All her skill set focuses on a strong AP build allowing you to be a bit flexible with other items. With high mobility and good use of skills, she is able to really influence the game.

I believe that using Annie to her maximum will involve a high level of team work in bottom or mid lane. With mid lane you are able to pick to assist top or bottom as soon as you can push you lane (when others would normally be recalling to buy items after a kill or strong push/counter). With bottom you are a bit more limited but that is made up for being in the very unique placement of RIGHT NEXT TO ONE OF THE MOST IMPORTANT CHARACTERS IN THE GAME! ADC is VITAL to your ability to take down towers or to win team fights. The high physical damage will allow your high magic damage more flexibility simply because having a high threat on both end is harder to counter with resistance. As an APC your focus is kills and objectives like Dragon.

The items that I find always helpful no matter what are chalice and Hourglass. The AP gain and ability to wait for skill refreshment without taking damage (or just not die when you flash+tibbers the other team for a hard engagement) make those two items more core then anything IMHO. Other people will say that DFG (deathfire grasp) is so core that to suggest playing a game without it is a folly but I say on to thee to travel to the top of the mountain before saying how fresh the water is! (IE. YOU CAN DO IT!) DFG, does give you a great edge and allows your stun locks to seriously kill the hell out of everything but honestly with a high AP build you kill the hell out of everything so much that DFG is just an item that you don't click on enough (Love it on high powered tanks though. Always exceptions) To me,

Armor and Magic Resist runes are very helpful for almost all heroes so I would recommend getting rank 1 until you get to level 20 and can unlock rank 3 runes.

Last hits with Q
Mana refund, Builds stacks, Scares other team

Annie is very hard to jungle with standard item sets but if you have ganked and have tibbers out it is a good idea to consider getting a buff

Sometimes you can use Soul's high AP stacks to gain enough buying power to do things like get the healing item, and deathfire to target down a high armor tank and add support even as AP carry... which is hill-larry-us!