Timing Flash Frost with Frostbite can lead to devastating combinations.
Anivia is extremely reliant on Mana for Glacial Storm. Try getting items with Mana or going for a Crest of the Ancient Golem buff on Summoner's Rift.
It can be very difficult for enemy champions to kill her egg early in game. Seize the advantage by playing aggressively.
Try to gank Anivia when she's laning. With multiple people, it is easier to ensure that her egg dies.
If you're playing a ranged champion, stay far enough away from Anivia so you can dodge Flash Frost more easily.
Try to fight Anivia in the lanes. In the jungle she can block pathways with lower ranks of Crystallize.