For Anivia i like to go for damage as much as possible because if you know all of her little tricks you can get away before dieing in almost any occasion.

Anivia's Tricks
With anivias flash frost skill you can set it off by pressing Q when the ball of ice is where you want it to be effectively stunning an opponent for 1 second (enough time to get away) this is a good tactic to initiate ganks, chase down enemies and counter people like Eve and twitch.

You can also do a funny trick called eggport. This is when you make use of the teleport summoner skill and activate teleport when you are on low health and have your egg passive ready, if you activate teleport and click on any turret in your minimap you can set anivia to teleport even when you're in the egg. getting away form enemies and gaining something to lol about.

also with her frost bite skill, doing double damage to anything that is chilled (turrets/inhibitors/nexus's dont work) it is a great bit of knowledge to learn that flash frost AND glacial storm are great ways to chill your opponent, i like to stun them and then use the storm to stack the chill on them (i find you do more damage this way anyway)

her crystalise skill in my opinion is now only used to defend yourself from melee dd's and help trap opponents in for an easy gank. you used to be able to cast this onto an opponent and freeze them into the wall, but this has been patched now sadly.

Also with your frost bits skill, if you time it right you can hit people twice with one spell, stunning them, chilling them and damaging them for usually about 400 dmg.

Early Game
Early game, weather going mid or top/bottom, its always best to follow this build and get movement speed first, anivia being one of the squisshyest (not a word) champions on LoL will need movement speed to be able to attack and retreat well.
if you however end up having a bad earlygame, swap the second archangels staff for a rod of ages.

Mid game
Mid game you should have all your spells but crystalise and be doing ALOT of damage to all enemy champions. (bancshee's veil is an easy item to surpass since it only stops one bit of damage in your glacial storm, which damages them 4 times in 1 second)

End Game

Anivia at the end game should never go solo unless you've mastered the eggport technique, and then only to do small ninja attacks like killing a turret whats on 500 health (at lvl 18 depending on the time you've been playing you should have about 800-1300 AP at the end of this, reflecting on your physical damage to turrets and inhibitors doing about 300 damage a hit!)
stay in a group, and initiate attacks with fros bites stun and a quick finish with your frostbite. simply do damage and run.



Additional comments

some people may say that getting getting a mejais soulstealer is a good idea for stacking AP but with these items you can get far more than mejais

Use zhonya's rings unique passive to assist in eggporting

when you go back to finish off your second Archangels staff (unless you have to go back) make sure you have enough money to get a tear of the goddess in addition to finishing it off, this helps you stack mana and therefore stak alot of AP.

survivability for anivia in this build depends entirely on your playstyle, don't be too offensive and be weary of ganks, in most cases try to shoot your flash frost, run towards them and when it hits instantly chain frostbite, then run away. i've won countless 1v1's like this so it is very effective.

Thanks for reading, and i'll welcome fyour feedback :)