Well there are many reasons I could personally think of when it comes to playing Anivia, for example
Anivia is a ranged mage with one of the highest bursts in the game even early game you will see your your health bars dropping by hundred every ten seconds of your life when facing one until you just have no more health and get first blooded, TEEHEE!!
Instead you wanna be the one getting First Blood, not getting First Blooded.
So I can make up just about any reason that you should play Anivia, including the fact because that she is a bird.
But if your looking for someone with a pretty high damage output and some nasty sustain and survivability, Anivia, is your girl.

Pros

*Hardcore CC.
*Great Farming with Glacial Storm.
*Very Effective both Early and Late Game.
*Easy First Blood with all of her basic abilities.
*Great bating tool with her passive, Rebirth.
*One of the best Mid Laner's in the game.

Cons

*Very Low Mobility early game.
*Can not easily escape ganks.
*Very High Mana Costs and CoolDowns.
*Egg can easily be fried if its not hugging any turrets.
*She is technically countered by any melee ap assassins with a gap closer; Shunpo Shadow Dance etc.

For Summoner Spells I take D: Ignite and F: Flash.
If you are low level then the following spells should suffice,
Level 1: D: Heal and F: Ghost
Level 2-5: D: Teleport/Cleanse and F: Ghost
Level 6-7: D: Barrier F: Ghost
Level 8-11 D: Ignite F: Ghost
Level 12-30 D: Ignite F: Flash
SO there you have it these are the summoner spells you should use depending on your level
other options such as Smite, Garrison, Clairvoyance and revive are not an option.

So why do you get the spells you do:
Well with any other guide you read for mid players they will say; "get summoner spells Flash and Ignite" or the other way around it really doesn't matter, however I get Ignite on D and Flash on F because F-lash.

Ignite:
So yeah like in every guide I get Ignite for further champion killing

Flash:
And of course I get Flash for quick escapes over walls or to cover more distance and mobility.

Why should you, under any circumstances get the spells I told you not to get.

Smite:
As you all know Smite is the summoner spell used by all jungler's so they can kill Blue/Red easily without dying. And as you all knew this spell is not to be used by Anivia for as most of you SHOULD know Anivia is not a jungler.

Garrison:
This is not a dominion guide and not many players use garrison in the first place, there are not many Dominion guides in the second place. Really you also shouldn't play Anivia in Dominion, Twisted Treeline or Howling Abyss (Btw if you're playing an ARAM and you own Anivia or she is free and you get her randomly picked you better be able to reroll or your ****ed and you might as well tell your team its GG). Anivia is an AP Carry that only plays on Summoner's Rift in the Middle Lane.

Clairvoyance:
This is a summoner spell that is to be used by Support champs and no one else. It is a waste of a summoner slot and a way better option would be to use Ignite for further champion killing, or Flash for further escape tactics.

Revive:
No Champion that exists would be very good with Revive you might have seen a few Kog'Maw's or Zyra's use this spell but let me tell you they are just trolling and playing with your mind.

Runes:

For Runes I take Greater Marks of Magic Penetration, Seals of Ability Power, Greater Glyphs of Scaling Mana Regen and Quints of Movement Speed. Here's Why:

Greater Marks of Magic Penetration:
I get these Marks because thanks to the Items Anivia buys she already has enough magic penetration to penetrate the base magic resist of any Champ in the League of Legends. Plus the bonus amount of your runes will just give you the opportunity to crush your opponents.

Seals of Ability Power:
The Seals of Ability Power will grant you a fair amount of AP before you get your first AP item. Which will just make you a complete bully in lane thanks to your runes and masteries.
Now I don't recommended flat out AP for your Runes but it does work its just that you will survive better with runes I recommend on the other hand I really don't mind you getting flat AP runes as it works just as well. Thing is you already have a lot of AP from both your masteries and Items.

Glyphs of Scaling Mana Regeneration
Thanks to your masteries you should already have a lot of mana regen and mana from your items (also a little mana regen from Tearess).
But this way you wouldn't go oom (out of mana) anymore and you won't have to disturb your jungler whilst he's stealing your blue!

Another good option would be flat AP Runes, but you should never bother wasting too much IP on just Runes and Rune Pages.

For Masteries I take:
Offense: Defense: Utility:
21 0 9
Utilizing all the AP in offense. Whilst gaining bonus Mana Regeneration , Movement Speed, Improved Recall, Cooldown Reduction, and Neutral Monster Buff Duration's in Utility.

Why do I do this:
Because the small boosts you get in Utility will have you thanking me later and with extra AP in offense you'll have like 50 AP (Inclusive your Runes) at the start of the game without any AP Items and you will have First Blood in no time unless some *** who is better claims it before you which won't always happen.

Items:

Why do you purchase the items you do:
First of all if you haven't looked at the stats up top you have decent defenses and you have a pretty big amount of AP at your service.

Defensive Items:

Guardian Angel:
Bonus Armor and MR plus a bonus revive and you already have Anivia's passive at your disposal so that's not a half bad idea either if you want to piss your opponents off.

Warmog's Armor:
Having a thousand bonus health really decreases the chances of people bursting you down really fast with True Damage abilities such as Darius's Noxian Guillotine and Cho'Gath's Feast.

Rylai's Crystal Scepter:
500 Health and 80 additional AP is very useful for Anivia considering she has the lowest bas health in the game (True Fact). This is because of her passive although it is not a certain or reassured revive, she is still given the lowest amount of health in the game despite Aatrox's or Zac's Passive.

Athene's Unholy Grail:
20% CDR 20 MR and 60 AP plus a big amount if mana regen will help you out a lot, you can trust me on that depending on you situation this item is not the worst option.