Anivia's Skill Max Order: E → Q → W
Level 1: Hide in top bush and conduct Quinn Cheese. Wait for passive to mark her, then Auto-E-Auto to consume to harrier procs. Anivia will retaliate with her Q. Quinn's E is perfect to dodge her Q since Anivia Q lands at a target location, so you can essentially use your E to juke her Q (eventually Anivia will pick up on this, but the first trade she will likely have no idea). If she misses her Q, run her down with as many autos as you can.
Level 2: Anivia will take E, which doesn't change much. Her E only hurts if she lands Q first, so the goal of this lane is to dodge Anivia Qs as much as you can. If she lands Q-E trade on you, instantly vault her and run her down because she can't retaliate trade after her abilities are down.
Level 3: Anivia has W now. Her W is good to help land Q, but it isn't problematic before level 6. After you crash wave to her tower, let the wave push back to you so Anivia is over extending. This is good because if Anivia ever misses a Q, you can run her down all the way to her tower and burn her flash/ kill her. This is why mages suck top lane, especially against Quinn, because the lane is long and you can run them down when they push.
Level 6: Your E cancels Anivia R and puts it on full cooldown. If she Rs, I would instantly E her, auto to consume harrier proc, then use the amplified movespeed to juke her following Q. She can recast R after 1 second, so try to be quick and escape when you vault her. Stay close to Anivia because she wants to poke you with range, but if you remain close, you can bully her hard and force her to take bad trades. She's really squishy, so one bad trade could become lethal for her.
Anivia wins if she keeps ranged advantage on you, but like all mage matchups, top lane is their weakness.
Use E to dodge Anivia Q if she ever launches it, and if you do this, she is useless because her E does no damage early game without her Q to empower it.
Stay in E range of Anivia always so that you can E her if she launches Q.
Anivia wins if you play long distance because her range is longer than yours.
Post 6, use E to cancel her R if she uses it.
This is good to mitigate the slow that it provides.
What I like to do is freeze wave in front of my tower, and this puts Anivia in a tough spot.
If she walks up to break freeze, you instantly engage on her and run her down since you are faster than her.
With top lane being so long, you can chase her down all the way to her tower and kill her since the lane is long from her tower.