So, this is my first guide and on Anivia, one of my favourite champions. I accept constructive criticism, but please no hate. I don't play much ranked, but have played over 250 games with Anivia. Please note that I didn't add boots or any health items/runes to this build since this is a build for more damage and less survivability.

Summoner Spell choice: I chose Clarity because of how much of a mana eater Anivia can be early on, even with a few mana items, you would be drained of mana for a good amount of time. Though, if you don't want to bring Clarity, you should bring Ignite, since Blue Buff can also help you with mana problems. Flash is for getaways, but since boots aren't a part of the build, you may want to use this over walls instead, especially if the enemy team has a lot of cc.Runes: Just full on AP runes. It wont help with sustainability, but with any mana item with Clarity will help you with your mana, but nothing can really help with your health, so back when you can, if you don't have Rebirth.

For the skill order, you want to get Flash Frost at levels 1, 3, 7, 8, and 10. Also, you would want to get your Frostbite at levels 2,4, 9, 13, and 14. For Crystallize, you want to get it at levels 5, 12, 15, 17, and 18. As for your Glacial Storm, levels 6, 11, and 16 will do. You want to try to max Flash Frost, but adding some points to Frostbite along the way. The reason is because you can deal a significant amount of damage and deal even more damage with your Frostbite, which does double the damage if the opponent is chilled (Your Flash Frost and Glacial Storm apply chill). Your wall is what you want to max last, unless you want more of a support role and separate teams. One thing to mention isAnivia's wall is somewhat glitched, letting champions pass through them at times!
Skill Tips: *Each different colour represents a different skill tip* While Laning, you want to try to use your Q through minions to get farm, even if you miss, it'll help you push. One thing to note: Anivia's Q can be detonated so if you think your Q will miss, detonate it at the mage minions.
Anivia's combo is pretty easy to land and master. Her combo is Flash Frost, Frostbite, Crystallize, and then Glacial Storm. In the beginning, Anivia can kill, or harass quite easily at level 2. Her combo then, is Flash Frost, Frostbite, followed by an auto-attack.
What is useful for pushing lanes, is her ult. It has a lot of CC and can mow down minions in seconds. Though, one drawback is that it is a mana drainer. What you can do is use your Flash Frost first, then Glacial Storm, while leaving the minions at low health to keep your mana from being drained.
If you have Lich Bane, you can use your Glacial Storm at the tower to take out minions and just attack with Lich Bane, and when it can be activated again, use your Flash Frost to activate your Lich Bane again while taking down minions.
One of Anivia's great skills is baiting. With her Rebirth and some teammates/turrets, you can drag the enemy champion to the rest of your team, reducing you to egg, giving your team a kill, and making the enemy rage.
For Anivia's Flash Frost, you can deal double the damage if you time it right by letting Flash Frost past through the champion, then detonate it.
Also, note that this build doesn't have boots. I intentionally did this since this is a full damage on build and would be a little hard to get away. If the enemy team has a lot of CC, then use your flash over a wall and block off a way that they can enter from.
Since Anivia's Q is quite slow, you may want to try tracking their movements and fire where you think they may go. What is also useful is waiting in a bush and activate Flash Frost to catch them on surprise and may get you a kill.

*Different colours represent different parts of the game as when you should get these items*
What you want to get when you start up is a Doran's Ring, a Health Potion and a Mana Potion. You may be able to stay in lane for a little longer and can deal a great amount of damage.
Next up are the Tear of the Goddess> Philosopher's Stone> Blasting Wand. What you want first is the Tear of the Goddess for some extra mana, sustainability, and to charge up for Seraphs Embrace when you get Archangel's Staff. Get a Philosopher's Stone to build up to Eleisa's Miracle and for some extra gold in between the time. Next is a Blasting Wand for some AP and to build up to Rabadon's Deathcap, or Archangel's Staff.
On your way to mid game,get an Eleisa's Miracle for the cooldown of clarity and sustainability. Through Mid-Game, you should get Archangel's Staff> Rabadon's Deathcap> Chalice of Harmony> Fiendish Codex> Liandry's Torment. Archangel's Staff is a great mana item and can give you extra AP for the mana you have. Rabadon's Deathcap is one of the most useful AP items in the game since it gives a great amount of AP from your current AP. Chalice of Harmony is to build up to Athene's Unholy Grail and Fiendish Codex for Deathfire Grasp. What you want next is to build, and get a Liandry's Torment since it will help you deal a lot of damage with your Glacial Storm.
Late Game, you would want Athene's Unholy Grail> Deathfire Grasp> Lich Bane. Athene's Unholy Grail is to help you stay in Teamfights since you gain 12% of your mana if you get a kill or assist. Next is Deathfire Grasp for CD. If you remember to use it's active, you can deal 15% of the enemy Champion's Health and amplifies magic damage for 4 seconds, making you a deadly force. Lich Bane is for pushing lanes since the spellblade depends on 75% of your AP+50, which can help in both fights and lane pushing.
Now the core items. The core items are: Seraph's Embrace/ Archangel's Staff, Rabadon's Deathcap, Athene's Unholy Grail, Liandry's Torment, Lich Bane, and Deathfire Grasp.

Through laning phase, you basically just want to shoot Flash Frost through minions and kit the enemy champion, or to get some farm. Since Anivia's Q is slow, you have to predict the enemy's movement. Even if you miss, you can get some farm from it. The enemy may not notice the Flash Frost through the minions either, letting you hit the stun, and harass. Also, when a teammate has called MIA or you suspect a gank, you can use Crystallize as a 5-second-ward. To farm, you should use your abilities instead of your auto-attack. Anivia has quite a weird auto attack-animation so you'd be best off using Flash Frost and Glacial Storm.

Pros:
Deals High Burst Damage
Okay amount of sustainability
Able to dominate early game
Great Pushing Potential
A lot CC
Great bait
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Cons:
Slow like snow
Low Health
Glitchy Wall
No Protection

Overall, this build is more in the offensive side than defensive. This build is focused mainly on damage so you wouldn't have much defenses. You would do better if you know most of Anivia's mechanics, know your way in dodging skills and have a good amount of skill for your wall.