This is the build that I've settled on for Anivia the Cryophoenix. This simple build focuses ap and mana regen. By the time you've reached level 11 if you've farmed well your ult can be set down in a team fight for an almost infinite amount of time. Then by level 18 you're the focus of each team fight because of the insane amount of damage you deal with your ult alone.

Early game start off with flash frost and take mid solo. Farm minions and harass the other champion as much as possible to keep them from farming and to possibly get an early kill. Push as far forward as you can when you need to b so that you don't have to use teleport to get back quickly. Once you hit level six if you haven't already gotten a kill let them push and chase you to your turret where you can throw up a wall to prevent them escaping and then lay down your ult and stun then frostbite for an easy kill with help from the turret.

Once you hit level ten or eleven it's time for the blue mana buff and ganking. Once your turret is secure mid go for the blue buff and go gank either top or bot whichever one needs more help. Swoop in for two easy kills or assists by laying down a wall and then ult and push the lane for towers. With the blue buff you should be able to keep your ult down infinitely and take out minion waves while warding off champions in order to take down turrets with ease.

You are the team fight difference maker because you can lay down your ult and deal almost half of all the enemy champion's health before they realize they've lost it and your team has killed at least three if not all of them.

Your Item order should go something like this. Start off with a mana crystal and work towards tear of the goddess. Next go for the chalice as that will fix all of your mana issues. Next finish off your archangel's staff. Then use your best judgement as to whether to go for the deathcap or deathfire grasp. Deathfire grasp has a fantastic active that could prove useful for getting solo kills. Finally depending on how the game is going either get a void staff for the extra ap and magic penetration or for lols you can build a Guardian Angel to frustrate your opponents who by the time you get it should be focusing you in team fights. This is the build that has won me an insane amount of matches. Enjoy and have fun.