DISCLAIMER: This guide was made in January of 2010. Items, champions, and the metagame have changed since then. Take this fact into account.
FF: Flash Frost
GS/Storm: Glacial Storm
Anivia is an amazing nuker, but is also amazing due to her CC and wall.
FF is NOT Anivia's main skill like so many think. It's just her initiating stun. Learn to aim this well and when to time popping it so it hits twice.
Purely amazing if you're good at placing it, but horribly evil if you suck. Learn this well, or you'll be a hated Anivia. Consider using it to make sure your FF hits the target.
Anivia's main skill and damage. Be sure that they're frosted before using it, though, or it's mostly worthless.
Early on, save Glacial Storm for griefing along with Frostbite and for Golem. Don't be stupid and waste all your mana by using it on creeps. You can do that late game.
Anivia is the queen of solo-mid. Unless you have Twisted Fate on your team, tell your teammate(s) to screw off if they want solo-mid.
Don't waste your mana on spells until level 2 or 3. Just farm minions until then. As soon as you get to level 2 or 3, start randomly stunning --> frosting the enemy, then backing away. Keep this up until you can kill them (don't forget Ignite). Be careful not to waste all your mana in doing this, in case of ganks.
You should have your Mejai's and be dominating the map. Crush the enemies with frozen death. If you are unsure whether you're good enough to be ganking yet, wait until you have your Rod of Ages.
Just like any other caster, you will be Death incarnate or a worthless squishy by now. If the former, just carry your team to victory with your horribly amazing damage.