Anivia is a very fun but hard champion to play, especially in the early game due to the amount of skill shots that need to be hit to do damage before level 6 and due to how squishy she is until you get Rod of Ages. I highly recommend starting Q and not using any mana until you have a point in E which is going to be the skill that does the most damage.

[Thunderlord's Decree] is so over powered right now and if you can get the enemy to stay in your ult for 3 procs its activates. So just drop your ult in a team fight and get the proc on all of them.

Farming can be very hard with Anivia due to her low ad and slow movement of auto attacks, but if you do not do amazing early game with farming her middle and late game will deffinately make up for the loss in the early game. Although the better farm you have the faster you can get the first two items giving you a huge advantage early game.

Anivia's skill sequence is quite difficult to pull off early game due to her low mobility and slow ability speed.
It is pretty simple though.
Early game: Q-Q (to proc stun)-E
Mid/Late game : R-W(on far side of ult)-Q-E

Flash is an obvious choice for Anivia again due to the low mobility and lack of escapes.
But Teleport is what a lot of Anivia players tend to forget about. When you are about to be egg'd teleport to a tower and the egg will continue with the teleport. Although it can be interrupted by any knock up, knock back, pull, silence, or suppress so when playing ranked find out if the enemy team has these in your lane and if they do take ignite.

Pros:
High Damage
Great team fights
Can completely cut enemies off from their team or yours with W
Great wave clear

Cons:
Slow movement speed
Difficulty to execute W
Combo can be hard to hit early game