I'm new to Anivia myself and I myself am not a professional player. I could not really find a guide that could point me the right direction of playing Anivia so I decided to write one as a guiding post for those who want to play her. Anivia herself is a champion that is most deadliest in teamfights as oppose to being on the field alone. This guide is made for sitting back and farming until when the teamfight phase begins, then Anivia can go all out cutting people off or splitting up their teams.

This is my build for masteries, you can choose to follow it or not, but I choose to pick a more resilent Anivia as she is more useful when she is alive then not. That and it will improve the chances of her surviving in her passive egg form for a second wind at a chance to destroy her enemies.

I have chosen (3)Quintessence of Fortitude because anivia is naturally very squishy in terms of health, that being said, the egg doesn't really help in your surivival if you have low health. Having these Quints will allow you to harass more with your auto attacks early game as well as give you better survivalbility in your egg form.

-(9)Marks of Insight, is pretty much self explanatory, in the long runs, it will help you with your magic damage, as any good player would do when playing against spellcasters will build magic resistance.

-(9)Seals of Clarity, Anivia is a pretty mana starved beast if not you don't control your mana usage, I find myself at a loss of mana to help teamfights or finish of enemies due to lack of experience. To counteract this, I use mana regen/lvl coupled with mana regen items to ensure I always have something to throw when need be.

-(9)Glyphs of Force, This itself is to help Anivia do damage as she progresses in level. I find that if you build mana regen items, you often lack damage you need to remain as a threat to your enemies. What these glyphs here does is to allow you to do decent amount of damage while building mana regen items in the beginning.

Summoner Skills, Flash/Ghost will be the escape spells for your champion.
Flash-This allows you to create a distance from ganks as well as focus fire if need be in team fights. This will also allow you to close gaps on enemies trying to escape from you.
Ghost-Allows you to reach teamfights much quicker as well as continue your combos of icy fury skills.

For Offensive Summoner skills, I would pick Ignite, good for finishing off healers/kills.
Exhaust is also a decent team fight as it allows your team to catch up if you are too far ahead as well as allow you to continue your combos on an enemy running away. It could also be used as a defensive skills as well when fighting against any hard core dps champions or for running away from them.

Utility, only two I can think of are heal and teleport. Like I said, Anvia is a champion deadliest when she is with her teammates. Being able to save a teammate or joining in a fight could ultimately change the flow of a game.

Do what you want, but I give priority to damage because walls won't do much if you aren't a threat regrouping from a bad position. Walls at most should give you a second or two leeway to breathe before you start shooting spells again at your enemy. At level 1 walls if position properly can cut off most paths. Higher level walling to me is for offensive team fights skills best saved for late game.

This will be it for now until I figure out how this website editing works. I hope you find this guide viable to your games. I will continue to play Anvia and update this guide for improvements when I see fit. In lanes it is just best to poke enemies with your auto attack and save your flash frost as a defensive measure early game before level 6. No champion can carry on their alone, teamwork matters.