Well im going to keep this brief.

Starting you should get Flash frost first, being your long range ability and stun you can poke and take off a decent amount of damage early game when landing this skill shot perfectly.
what you will want to do is position your self so if you ability misses the champ that it hits minions so its not a complete waste. (remember your flashfrost can be activated twice, it deals damage when passing through an enemy but also if activated a second time it will detonate dealing damage again and stunning them. so throw your Q and detonate it just as it hits your enemy maximising your damage. )

Early game you should be playing semi aggressive (your passive saving you from death quite easily), throwing out your stuns every now and then, if you land a successful stun go in for frostbite if done correctly, your flashfrost should hit twice and your frostbite should deal
double damage taking a solid chunk out of their HP.

continue to do this and make them recall as much as you can lowering their level and cc, once you reach level 6 your ult is up, this ult can help you farm very fast, if you keep harassing the opponent they will be over run with creeps, Beware of the enemy jungler so don't push to far ahead to much, it takes about 200 mana to kill a group of creeps so with your chalice of harmony your mana will be back to 200 before the creep group returns (kill the group and wait back in the middle away from danger and by the time your creeps die the new enemy creeps will be back to feed you up again),

if you continue to do this your creep count will increase rapidly also your gold will pile up before you know it, just continue to cc and you will out power your opponent in no time.

Some burst sequences I use

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Burst one - Q-E-R-W

burst two - R-E-Q-W-E

Burst one will be most used ganking/killing, where as burst two is best used in team fights lowering the HP of opponents while slowing them with your R, dealing double damage with E, followed by your stun to keep them in your R, throw up your wall to prevent them from running and by this time your E will be up again resulting in mass damage to the whole enemy team.

If your having trouble landing your stun don't forget that your R also applies "Chill", so drop your R and follow it up with your E to deal double damage.

In conclusion, Landing Q perfectly is a must with Anivia, and following up your Q and R with your E. Other then that the rest is up to you to learn her abilities ranges and such
, little tricks like checking bushes with your W and things like that will follow naturally.

Thank you for taking the time to read my guild, it is the first guide I have done and would appreciate some feed back on how to improve my guild maker and/or this anivia guide, I am open to all comments and feed back.

P.S look at the notes about item sets for details about the items and why I chose them.
Thanks.