A good offense is the best defense... and a great offense is just fun.
Anivia has incredible burst damage, starting from level 2, which last I checked, counts as early game. Basically, what I do is harass the enemy with auto-attacks at level 1 (running into bush or behind your minions straight after the attack to avoid aggro) which can down about a bar of the enemies health before level 2. At level 2, throw a FF and FB and if you've done it right, that's about 2 bars of health.
By this point, almost any champ will have less than half health, unless you're laned against an tanky champ. Seeing that your combo takes out nearly a half of their health (more than half at level 3), most players are wary of you. If they aren't, unleash your combo, throw in an ignite if you have one and grab an early first blood.
Besides that, once they start getting scared of you, hide in bush a lot, and just come out for last-hits. As soon as you run into bush most champs tend to run 5 paces back, and still be a nice FF target. If they're brave, stun them as they run into the bush and break down their health again.