Ok for starters this is my first build so i hope you enjoy!
Anivia is large burst damage caster who if played right can burst down most characters.
She is primarily played in mid which is her optimum position.
In this guide i will develop how to play her and what you should build.
Pros:
Very strong Burst
A free Revive!
Strong all game
Heaps of CC (stun, slow, WALL)

Cons:
Slow movement speed.
Very squishy
slow attack animation.

Firstly Runes.
i take
Marks: Magic Pen x 9
Seals: Flat AP x 9
Glyphs: Flat AP x 9
Quintessences: Flat AP x 9
This setup allows me to have early game dominance.
it gives 29 AP and 8.5 MP which will allow u to do tons of damage early, which is what we want.

I use a 21/0/9 setup.
maxing out all of the damage in offence then putting points in mana regen, the Summoner Spell point, movement speed and spell vamp.
once again trying to get a lot of early game damage.
with these runes + masteries i start the game off with 36 AP.

as for farming Anivia's Attack animation is extremely slow, which makes last hitting rather hard early game. there is no easy way to do it, so you will just have to get used to the slow animation :)
early game its all about timing, it may take a little while but once you get used to the slow attack animation it gets easy.
once you hit level six its the easiest way to farm.
lay down your R and let it do the job.

I take Flash and teleport for spells.
this allows me great mobility all game.
Teleport incase u need to get in lane quickly or if u see an opportunity to gank another lane.
Flash can be used to chase down enemies or ever get out of a sticky situation.

Q-E-E-Q-E-R-E-Q-E-Q-R-W-Q-W-W-R-W-W (Q-R>E>Q>W)

Firstly you want to start off with you Q ability, which is your stun.
This ability first is great, if u happen to either gank or get ganked you can stun you enemies and either kill them or run easily. also if you time it right you can get off a double damage hit, by reactivating the spell to stun just after it has passed the enemy. this early will get most champions to about 3/4 health.
Next you want to max out Your E, this is your double damage ability.
This is your Burst Ability. If you enemy is Frosted and this hits, BAM, they take double damage. Once you hit level two you should be able to burst most champions down to about 2/5 health with your Q and E combo.
Your wall can be useful, but i tend not to use it early game as i think your damage is more important. however can be used late game to separate the enemy team.
Your Ulti is amazing, a rather big AOE that does Damage over time. you want to get this as much as possible.

As for combos this should be what u try and do every time.
Open out with your R, it slows the enemy for you as well as frosting them.
once your Ulti has hit them you will want to toss out your stun and try and hit it so it does the double damage effect, then lastly for the killing blow you use your E.
If you have done this correctly you will almost kill your target.

Anivia's spells tend to cost a fair amount early game. this is why it is a must to start off with a Mana Crystal. this allows u to use your spells more freely and not have to worry so much about missing the occasional ability. also u want to have two health potions just in case u happen to fall low. as soon as you have the money for a catalyst you should B and buy it. this item is amazing early game for any caster particularly Anivia. it gives more mana and health plus the passive which is awesome if you are low on mana or hp and you just can't kill the enemy, u level up and bang u have enough mana and hp to finish the job. next u want to get yourself some boots as Anivia is one of the slowest champions in the game u will need the speed. next you want to head for a Rod of Ages. this item is all-round great. it give you heaps of AP, mana and hp. which is everything you need as a squishy caster. also u gain even more as time goes on. Note: be sure to buy 2 or more wards if the enemy team has a jungler and place them on either side of the lane. this is to make sure u do not get ganked and keeps u alive. 150 gold is a small price for your life. once u have your Rod of Ages you would want to finish off your boots, into Sorcerers shoes. Anivia has a lot of mana issues so you will want to grab a tear of the goddess, this will allow you to use you spells more often and allow you to use your ult for longer periods of time. at this point you will wanting to pile up the damage. so a Rabadons Deathcap is the perfect item to get next. the 140 AP plus the bonus 30% will give you tons of damage. At this point you should be nuking down almost anyone on their team. the next item you want to get is an Archangel staff, this is like the rest gives heaps of ap and mana. Anivia is a very Squish champion and late game your egg cannot save you much so you will want to start and build some survivability.
There are some options:
I tend to use a Banshees Veil, which with the health and magic resist gives heaps of survivability against their casters, also the spell shield is a life saver.
I also get a Zhonyas Hourglass, this allows me to be more tanky against their carry and gives more ap. also who doesn't want to be a golden statue?

Other Options:
Void staff, if they are building a lot of magic resist build this or a abyssal Sceptre.
also you could even build another Archangels staff.
There are many more options available, test them out!

1. Using your wall. this at times can be very effective, however it can be used so it does more harm then good. be sure to be careful when using it around enemies with an inbuild flash.
E.G. you are chasing down a tryndamere. you use your wall to try and keep him near you (blocking off the entire pathway) he then spins over it therefore making you stuck behind your own wall for 1-3 seconds. lost kill.
also when using it when in a team fight, becareful not to block out a teamate or block him in so he dies.
However it can be used very well to keep enemies in your ultimate or even to get a better show at them with your Q.
2. try and make sure that you activate your Q just after it passes through them.

Hope you enjoyed my guide on how to play anivia.
hope you guys have fun dominating people in the battle field :P
also please vote up and comment with suggestions. as it is my first guide i have no idea how to do much so apologies for the lack of pictures etc.
Thanks Morty
LoL username: morty1