Here is the my build for anivia or what I like to call her Articuno. I'm gonna talk about the runes and masteries first.
Runes: 9 Greater Mark of Insight, 9 Greater seal of Replenishment, 9 Greater Glyph of Potency, and 3 Greater Quintessence of Potency.
The Masteries are going to look like 9-0-21. You are going to want to take one of those points in the neutral buff thing because it will make the blue buff stay on you a little longer.

Items: Want to start the game off by buying a sapphire crystal, health pot, and a mana pot. Right from the get go you are going to want to turn that sapphire crystal into a tear of the goddess. You want to build up your mana fast because she will start to chew through her mana when you get to lvl 6. Then after you get tear go ahead and get Sorcerer's shoes, then build up to rod, then deathcap, the archangels staff, then zhonyas hourglass (make sure to put that into your first item spot so you can use it) then after that you can go really anything with that last item, like void staff or Deathfire grasp if they have a bulky tank.

The summoner spells that i use with Anivia are Flash and Teleport. The reason i use flash is because it is a must with any ranged ap champion it can get you away from danger like that if you know how to use it correctly. Then the next summoner spell i take is teleport because i would be soloing mid most of the time cause she is an excellent mid champion, and the reason i use teleport is because it gets me back to my line faster then walking there and you can gank pretty well with it.
Some of the spells that would be alright with Anivia would be Ghost or maybe ignite, possibly clarity but if you have the runes and the right items then it would just be a waste of a spell.
Spells that i would never take: exhaust- you are not the tank to gtfo with this spell, rally-screw rally!, Smite- ya have you ever seen a jungle anivia... ya thats right NOOO! So that means don't even look at this spell.

Anivia is an exzcellent farmer, in my mind she is one of the best farmers! Her ult just chews through minions from the beginning, and what I like to do is combo the ult with sending flash frost through them to clear the minions faster for a faster lane push. So when a new minion wave pops up all you have to do is use your ult and send flash frost through and you are good.

Pros of anivia would be, she is an excellent farmer, she has alot of burst damage, she hits hard the whole game.
Cons of anivia would be, she is pretty squishy thats why i recommend rod of ages with her, she has a hard combo to get used to, and she doesnt have the longest range of a ranged ap champ.
Combo: The combo that you have to get good with is first you flash frost (make sure that it hits them and right after it hits them hit q again and it will stun them and do more damage), right after that go in an use Frostbite, this will do alot of damage because if they are chilled it does double the damage, then after that if you have ult, go ahead and use it then try to get the wall in front of them so they have to run around it and they will take alot of damage from your ult and by that time your frostbite or flash frost should be off cd.

Early game: so what you want to do is go ahead and take mid lane, and you are probably going to solo against another ranged ap or an ad carry (ashe, caitlyn). What you are going to want to do is start getting last hits on minions and try to land flash frost on the enemy champion, it will hit hard in the beginning, but just remember that it is pretty easy to dodge so don't get annoyed if you are not hitting as much as you wanted to. You will then get Frostbite and this is when you can start doing alot of damage to the enemy champion. Also remember that her passive is that she goes into an egg every 5 mins if she gets slain, so this allows you to be aggressive, but don't want to overextend. So you are going to just keep and try to harass the opponent. Once you get to lvl 6 then you are going to start farming pretty hard and your going to be pushing the lane pretty hard, but with that you also got to remember that you could get ganked easily from bot, top, or jungle so care!
Mid Game: Now you get to mid game you are going to want to grab blue as much as possible because it is SOOOO good when you have it! Her mana regens so quickly and you can basically have your ult on the whole time and it won't even go down which in my mind is pretty stinkin awesome! So what you are going to want to do is maybe put a ward up on blue so you can get it when it comes up, you can take it easily by yourself. By this time you will be chewing through minion waves and you can start ganking top or bot by going through river or you can teleport to bot or top and do it that way.
Late Game: Once you get to late game you will have alot of AP, you will have about 750 (I dont know why it says only 438 up there.) But you just got to make sure you stay back cause you are going to get focused down! And if they do that you will die fast. But what you are just going to do is try to get blue as much as possible, and just do as much damage as you can. Practice with placing your wall in perfect spots to try to single out some opponents that your team can jump on.

I would like for people to rate this build and comment on it. Thank you Summoners!