Hello there! It is my second build so graphics maybe won't look to good but trust me, I played Anivia around 200 times and i know what is good for her :)

REMEMBER WHEN PLAYING ANIVIA ALWAYS GO MID OR SOLO LANE NEVER GO 2v2 LANE.

BLUE BUFF IS NECESSARY NO ONE IN YOUR TEAM NEED IT MORE THAN AFTER 8 LEVEL YOU NEED TO HAVE IT EVERY SECOND IN GAME.

As you can see in first build I use Marks for magic penetration Seals for flat AP Glyphs for flat AP and Quints for flat AP. Why so much flat ap ? Because Anivia needs this 30 AP at start.
It grant you with good firepower and a big chance for first blood.

In second build I use Seals and Glyphs for AP per level, and Quints for flat AP.Why? Cause you need some AP at begin and Seals,Glyphs for AP per level brings you more in a couple of time.

No matter what ALWAYS begin with, don't pick Doran's Ring or Amplifying Tome, cause you really need to have this 200 mana and two healing potions

First Build

-To increase your mana over time I think it's best choice for first item, cause you will be gaining mana every time use ability. Later you won't have so much time to do it.
-I leave it without comment cause it's not necessary here :)
-A big boost for AP get it as fast as you can
-Some AP boost and also survival
-With your ammount of mana and Rabadon it will give you a big AP boost.(Tear of Goddes gained mana transfer to it)
-No one want to be squishy right ? :)
--``- + cooldown reduction and slowing nearby enemy attack speed

Second Build

-Grants you with easy leveling :)
-I leave it without comment cause it's not necessary here :)
-Usefull item if u gain kills or assist :) if u die a lot don't pick it and when you see that you won't get many stacks sell it and rush rabadon you will later replace it with void staff.
-Some AP boost and also survival
-A big boost for AP get it as fast as you can
-With your ammount of mana and Rabadon it will give you a big AP boost.(Tear of Goddes gained mana transfer to it)
--``- + cooldown reduction and slowing nearby enemy attack speed

Additionaly you can pick
When have not enough or too much gold always buy elixir of brilliance cause it's very usefull for you :)

Rebirth-, your epic passive. Yeah that is the skill that people call "OMG AGAIN THAT F**ING EGG SAVED HIM WHAT A LUCKER" Your beloved skill that will save your *** many times
Flash Frost- This skill can deal nice damage and stun enemies when at end of the range or when detonated by you. Use this skill to stun and chill your enemies and blow them with your Frostbite or just to run away from someone.
Crystalize- A wall of ice that block enemies or minions. Use to scan brushes and to block your enemies when chasing you or your allies it helps your team very much.
Frostbite-It is a damage skill which deal a double damage if target is chilled.This is your main damage skill, you should max it first. Use it only when target is chilled by your Flash Frost,Glacial Storm or Nunu's or Ashe skills
Glacial Storm-This is your ultimate.With it you have very good CC in combo with stun and wall. Use it to farm creeps and to slow,chill your enemies to land a powerfull frostbite or just to run away when chased.

Only these are usefull spells for Anivia. I recommend to use Flash and Ignire or Flash and Clarity cause it is the best way to play.

What to say... Play like in normal games but remember about wards.Have always these green eyes in brush on the left and brush on the right sight of mid lane. They'll be ganking you! You need to be prepared.When on solo lane get one in brush and second at river just before dragon/baron

+ Deal a lot of Damage
+ Very usefull for team
+ Unstoppable at Mid or solo lane
+ Good counter against all champions which not have silence (especially good vs annie)
- Squishy at beggining
- Might have problems with Malzahar or Kassadin (if they play good)

Use your Flash Frost to intiate a fight, to help your allies when they're chased.
Use your Crystalize to block enemy champions from killing your allies or use with Glacial Storm to keep them passing through it.
Use your Glacial Storm to kill enemy minions and to not let them out from somewhere.

Put your Glacial Storm on and use Flash Frost through minion wave to kill it almost instantly :)