Welcome to my first guide. If you're reading this I'll assume you have a decent grasp of how to play Anivia. If you don't, here are some basics you need to know. Anivia is a ranged AP carry. This means her spells are incredibly strong but her physical attacks and health are ****. Anivia relys on her chill effect from Glacial Storm and Flash Frost to get the most out of her damage so if you aren't good with combo champions you might want to pick another before you start QQing all over the place.

Magic pen marks just make sense, and I get the same for Quints. Get AP if you want, I personally don't.
Seals I go for clarity. Since you won't be getting mana regen from Items till later in the game it's good to have some.
Glyphs it's either Cd reduction per level or more magic pen

Standard AP dps champ rules apply for masteries, 9/0/21. I get the point in blink of an eye because I use flash, If u don't want to then don't.

Get Flash/Ignite. U can get ghost, clarity, teleport or exhaust if you want, these are all viable options

Priority is Glacial Storm (Ult), Frostbite (E), Flash Frost (Q), Crystallise (W). Get a point in crystallise at lvl 3 or so if u feel the need, If I need more defense I sometimes do. 1 in Flash Frost at lvl 1, no point in Frostbite yet because the double damage won't trigger without the chill effect from your Q or R.

Sapphire Crystal + Hp pot + mana pot.
1st trip back get catalyst unless u feel the need for speed, in which case get sorcerors shoes, or boots and ruby crystal if u don't have the money yet.
Get Sorcerors or Catalyst if you haven't and build into RoA.
Get Rabadons deathcap and watch the enemies die.
Rylai's.
Next Items are purely subjective. I prefer AA staff and Void staff if enemies are stacking MR, if not Zhonya's Hourglass.
I also find Lich bane a good item to get, once you have around 700 ap your auto attacks become a nuke.

OK. U want to own. U want to own goood. But how I hear you ask. 1st of all go mid. You can side lane, but you don't want to.

Anivia is squishy sure, but her burst damage and ridiculous passive mean you can harass like a ***** and then piss on your enemies shoes and they can't do **** about it. save mana early, but if you spot an opening don't be afraid to take the opportunity. Always be concious of your egg timer. The number of times I've run out on low health to make a noob tower dive me is incredible, and usually followed by much QQ spamming in chat about how it was on cd.

Early game just farm farm farm. if your enemy gets too close throw a flash frost their way, they'll either dodge and not attack you or they'll get hit twice and stunned and not attack you. By lvl 3 ur Q-E combo will do a whole mess of damage, so play a bit more aggressively. By level 6 you should have a kill. If not then try harder. Once you have, gank, and gank like a boss. Your combo should be Q-E-R, W is purely situational, you should know when to use it. If you miss Q then R before E for obvious reasons.

Late game your AP should be pretty crazy, so just keep doing what you're doing :P.

Just some little tips for you on your travels.

Crystallise gives Vision, so pop a wall in the bushes if ur afraid of a gank.
Flash Frost has good range, abuse the stun to help your team. Don't save it just for the Chill, your R does the same.

That's about it :P First guide so rate up out of pity!!! Jk. If u want to vote down comment so I can take your advice.