For skills Build QEEWEREQEQRQQWWWW or get W at lvl I usually get it at lvl Ȅ to aid in landing Flash frosts.
I like Flash and Clarity Flash is great for getting FB, escaping, kiting etc. Clarity keeps your mana up and allows you to avoid Mana items letting you focus more on damage.
Now as anivia I pretty much always take mid, I can beat just about anyone in mid, the champions you will have the hardest time against will be kat, and vlad. Against them its best to be very aggressive early game as you are stronger than they are. Try to bait them into A flash frost frostbite combo at lvl , by waiting until one of your creeps get low and launching FF at them when they are about to attack your creep. Auto attack them as much as possible, anivia has ƀ range and i usually do about half my damage in the lane from auto attacking, the boots help you with that a lot. Also try making them walk into a flash frost by walling behind them and launching it in the direction you think they will run Once you hit " the lane is pretty much yours, I usually push the hell out of the lane or zone my enemy. Try to get them in more Q E or R E Q combos. When you get the opportunity to gank top or bot do it, constant ganking of the lanes helps your team out a ton, the faster you get lanes pushed and towers down, the more gold your team earns and the better chance you have of winning.
Pros: Anivia has amazing burst damage and support for your team. A stun, slows, and a wall that if used correctly can completely turn a game around. It's pretty easy to keep your enemies at bay and away from you with her skill set. She is one of the best ap carries in the game hands down.
Cons: Flash frost is rather slow and makes up for half of anivia's burst damage sometimes tricky to land. Her movement speed is the slowest in the game, she gets wrecked by Yi, Olaf. Mana, anivia eats mana like a fat man at a buffet, you need to constantly be grabbing blue buff or have to back more often.
Working in the team
Pretty much you stay behind the team let your tank initiate then blow your load on the carries, try to wall off enemies to separate their team, making team fights an assured victory. Save Flash Frost to stun champions that have channeling spells such as Kat's ult, nunu's ult, fiddles etc.