Anivia mainly has a magician role, so if possible, it is best to take her on Central Street (Mid), or Upper Street (Top). If there is no other option, it is also viable to take her as Support, buying for her the Ancient Coin object instead of the Ring of Doran. As a Magician focused on Ability Power:


Early game
Start by activating Frozen Flash, and take advantage of it to harass your enemy, but without jumping for him, and keeping your distances. Focus mostly on farming (killing minions), and try not to go back to base until level 6.
At level two it unlocks Freezing Burn, and at level 3 it enhances it, at level 4 it unlocks Crystallize, and, from there, dedicate yourself to enhancing the skills in this order: Freezing Burn> Frozen Flash> Crystallize (at levels 6, 11, and 16, empowers ultimate ability, Glacial Storm)
Take advantage of his passive so as not to leave the line in the first few changes, but still try not to spend it until at least you reach level 6.
If you are forced to return to base before reaching level 6, analyze the reason, if it was due to lack of life buy Ruby Crystal, and if it was due to lack of mana, then better buy Sapphire Crystal, both with the idea of ​​making the Rod of Ages. However, you should prioritize the Tear of the Goddess, with the idea of ​​making the Staff of the Archangel, to collect accumulations as soon as possible.
At level 6 he returns to base and buys everything he needs because now the interesting thing comes.
Mid game

Now that you are level 6, you will realize that farming has become a child's game, simply cast Frost Storm so that you catch the maximum possible number of Minions in its range (do not waste it, if there are less than 5 enemies is not worth it.
If you catch an enemy, try to hit them with the following combo: Frost Flash (detonate to Stun), Frost Storm, Frost Burn, Crystallize, and, if you have time, another Frost Burn. If your enemy was with less than 50% life, it is possible that you kill him with this, or, at least, you will force him to return to base or hide under the tower, a moment that you can take advantage of to knock it down.
If they are ganking you, get under the tower and try to farm with your Ultimate from there, if an enemy throws himself under the tower and kills you, the passive will prevent them from giving him death, and see if he is able to kill you before the tower destroys him .
Late game

Anivia is the goddess of Teamfights, if you know how to use her. Start by rolling Glacial Storm where the enemies are and then roll Icy Flash or Icy Burn, depending on the occasion, if they try to escape, calculate the distance well and roll Crystallize to cut off their retreat, and, if you can take death, better, although it is attendance is not bad at all either.
Your passive can also be useful, because while your enemies focus on the egg like crazy, your teammates with a little luck can escape or, in the best case, take advantage of the focus and destroy them.
Remember that Anivia is what is considered "paper", that is, a lot of damage but very little life, so don't even think about throwing yourself into a teamfight.
General considerations

Against the dragon and Baron Nashor don't bother using Icy Flash, because they can't be Stunned. Instead, choose to hold Frost Storm for as long as possible while harassing him with Freezing Burn, and most of all, don't try to do it yourself, at least at first.
It is possible that your enemies are going to focus on you as soon as they see that you are wreaking havoc (something that with a little luck and if you are moderately decent playing it will not take you long), so do not hesitate to ask your jungle for help, or, in extreme cases, play under the rook and farm until you see a chance to climb your line.
If they are chasing you, the best thing you can do is flee into the jungle, and, once there, place the wall of Crystallize in the middle of you and your enemies, that will totally block the street and you will be able to gain precious seconds. Stunning him with Icy Flash can also be a good option to gain a bit of distance before placing the wall.