Anivia is by far the best game maker in League of Legends. She is capable of making or breaking any game. Experienced players love her, inexperienced players destroy her.
If you really want to master Anivia I suggest you read through this whole guide throuly as I will explain different play styles and tactics.

For runes i always use the same runebuild when playing Anivia. This is becouse this is the runes that fits my playstyle with her, you are free to mix it up a little if it suits you better ofcourse.

I go flat Magic Penetration Marks : Anivias spells hurt like hell, with spellpenetetration
earlygame you will basicly rip people to
shreds at lvl 5 together with sorceres shoes

I go flat Armor Seals : Some people might argue with me on this one, but
anivia is really squishy and vounerable
earlygame, ive found that the extra armor
has saved me in many situations.
Take for example you accidently allow yourself to get
ganged, with the armor seals you will escape with a
little hp left instead of loosing your egg over a
gang gank, as most junglers do flat damage.

I go Mana Regen Pr. Level Glyphs : While flat mana regen glyphs is nice to have, I have
found that for anivias case its better to
start out with a little less regen to gain
more while gaining levels. Anivia has manastruggle,
so around level 7 you will start to notice the effects
of having a little extra manaregen to bide your mana
with ;)

I go Movementspeed Quintz : Anivia is really slow, and being slow is being
targetable. So i up the movementspeed
to make her utility a little better

The masteries are pretty straight forward.

I go 9/0/21 Reaching for extra spellpenetration on offence while going for bonus experience extra mana and movementspeed on utility

You can aslo go 9/0/1 reachin for extra manaregen and improved flash over movementspeed.

I have tested a lot of different builds, and i found this one to be the most effective for all around play.

I dont like snowballing but there is nothing wrong with picking up a mejas soulstealer before picking up rod of ages if the game is going really good for you, the extra ap really cuts through.

I start with Mana Sapphire, then i go for tear of the goddes, wich is a must item on anivia.
I then go for Sorceres Boots, wich i find to be the best boots for Anivia with the bonus spellepenetration. I then move on to Catalyst the Protector for extra hp and mana + the nice regen effect on level up and building it in to Rod of Ages.

After this there are different paths you can take.

You can rush Zonyas Hourglass if the oposing team is focusing you.
Or you can go for Rabbadon's Deathcap. I suggest you get both, one now and one at the end of the build as huge packs of AP is anivias best food :D

You need one of these items, when you get either one ur around 25 minutes in to the game and Anivia really needs to build AP in mid/lategame.

I then go for more nuking power with Voidstaff(40% extra spellpen really makes a difference in damage), but you can also go for Lich bane here as the extra autoattacks will hit for 800 bonus damage when your build is complete, playstile preferences ofcourse.

As you see in the skill sequences i max out frostbite as fast as possible.
Frostbite is anivias main damage, it hits for double its damage as long as the oponent is chilled ( Flash frost and Glacial storm both applies chill effects).

I dont skill wall untill level 10, this is only my personal preference and you can skill it already at level 4 if your team is ganking and such. But wall normally isnt needed untill mid/lategame for setting up teamfights and preventing enemy gangs/escapes.

The 2 most effective summonerspells for anivia is Ghost for movementspeed and Flash for escapes.

You can aslo go for Ignite or Clairvoyance if your team needs, but FLASH is a MUST when playing this champion.

Pros : Extremly good damage, and a really good skillshot stun, also a slowing ultimate of
madness
Anivias Walls can easily change the game
Viable with any setup
has good CC and has an extra life.

Cons : Very squishy
Very slow
Her walls can ruin a game
Anivia is better mid than lanes
Depends on Bluebuff

Anivias strenght lies in the ability to hit with flashfrost - Frostbite combo, wich can be hard as it is a skillshot.

Flashfrost doesnt stun unless it hits the max range or gets activated before it. It moves cinda slow and is easly dodged.
You will need to practice this a bit before you get the hang of it. After you successfully stun your target toss in a frostbite and watch the enemy hp vanquish.

When laning this combo is both your attack and your defence. I dont reccoment spamming the abilityes whenever you can as you would want to keep your Flash Frost unused for most of the time as it will break a gang when you stun them and run away.

In a teamfight and Mid/lategame

use this skill rotation : R - E - Q while keeping W ready to wall them in or out.

With your ulti slowing them down and the nuke from Frostbite most players will try to get away, toss your Flash Frost after them and stun them inside the ulti, you can also wall them to keep them in even longer, when you hit your "E" skill the 2. time most champions will be dead.

When laning earlygame, you will struggle a lot against nukers, Annie and Leblanc being the worst 2 champions you possibly can meet at mid.

Watch your position all the time, if you are to far in front you are likely to get killed.

Anivia do the most damage ingame next after leblanc on single targets, but you are the most squishy champion in game and you will die if you get jumped by a stunner/silencer.

Anivias Walls can at the maxed out level separate everything, even the river around baron and dragon if placed correctly

When anivia dont have the bluebuff you go out of mana in about 10 secounds with ulti on.

Hope you like what I have made for you so far. I will continue to update this guide over the next few weeks.