It is generally recommended to time Rebirth while under an allied tower or while you have your allies for protection.Do not count on your egg too much when you are alone and not under a tower. Low-level enemies can kill you because of the huge armor + magic resistance reduction and long revive time. Try to be aggressive in teamfights so you can deal good damage; be focused and let your team do the rest while they start working on your egg.
Rebirth can be used to bait an enemy to commit to killing Anivia when your jungler ganks that lane.
Despite having Rebirth, Anivia is fairly vulnerable to other harassment. Heavy hits can severely cripple her. Try to stop harassment by stunning your opponent before they can combo you to win tradeoffs.
Both the damage from passing through and the detonation of Flash Frost remove a spell shield from Banshee's Veil. This means if you time it correctly and detonate Flash Frost right after it passed through an enemy with Banshee's Veil's spell shield, it will dissolve and the detonation will deal full damage, stunning the target.
During the laning phase, use your auto-attacks to hide harassment attempts. Stand within attack range of an enemy champion and auto-attack minions with Anivia. When the enemy doesn't expect it, launch a Flash Frost and charge straight at them; if you time it correctly, they will be chilled and stunned, and you'll be close enough to land a follow-up for tons of damage. This is an effective strategy because the start of many of Anivia's spell animations look similar to the beginning of her auto-attack animation, and this can often catch an enemy, expecting an auto-attack, off-guard.
Anivia'sFlash Frost explosion area of effect is slightly bigger than the initial projectile's area, so try to hit the opponent twice with the spell by detonating it after it passes through them.
Anivia's wall has a cast time, so be sure to cast it WELL in front of the opponent. Just a little in front will block you off and push them away.
In the jungle, the wall, Crystallize can completely block off a pathway. Allies can be blocked in/out as well so timing is essential.
In the jungle, Crystallize can completely block the Ancient Golem in its area. This allows you to kill it without taking any damage as it cannot reach melee range and attack.
Crystallize grants sight, you can cast it in a bush to safely check them.
Crystallize will grant assists if an enemy champion touches it before dying. Be careful of turrets targeting you if an enemy touches your wall.
It is also possible to disrupt channeled ultimates' animations by putting Crystallize directly on top of the opponent. If done correctly will shift the opponent from his/her position, canceling the animation, but not the ability. This is useful if Flash Frost is on cooldown.
Be aware though that only the animation is stopped, so the abilities effects are still in effect until they are hit by a hard crowd control effect or take action.
Glacial Storm is a great spell that can clear entire minion waves very quickly. Be sure to deactivate the spell when you don't need it as it will quickly drain your mana.
Glacial Storm is an ideal tower taking ability, assuming you have blue buff / high mana regen. You can drop your glacial storm on the wide gap of the turret where the minions come out. This gives a lot of enemy champions problems especially if they are melee due to them having to make the choice of walking through the storm and getting hit with a Flash Frost to Frostbite combo or try to make their way around the small areas left not covered by the glacial storm setting up for an easy FF/FB combo. The same concept can be used from enemies playing offensively and attacking your tower.
Crowd control effects that stop abilities from being cast, such as stuns or silences, immediately stop Glacial Storm from being channeled. If Anivia is not guaranteed to die within the duration of such crowd control effects, it may be wiser to save them in expectation of Glacial Storm to stop it from being channeled.