As a level 7 Anivia main, she's amazing when you learn her. Ever since when I first started league and saw they had a cryophoenix I was like "wow she looks cool!" Little did I know she'd grow to be my best and favorite champ.
Her 1-5 isn't awful. Last hitting is hard, but if you can surprise the laner with a q>e it does 1/2 their health. Eventually, you'll get the hang of what to do. Once you hit 6, Anivia is very hard to deal with in lane (Take teleport so your mana problems aren't as much of an obstacle)
Mid-late game, if you just pay attention to positioning, you'll be fine. I find that if you're ahead, you make plays (wall into q stuns). If even/behind, you stay back and wait for the enemy to engage. This is where you aim your q to go to the back line and stun the carries (they rarely ever watch for it) so their engage is wasted. This is generally 66% of how you'll play most teamfights.
Walls are what separate good Anivias from great Anivias. I still am learning how to consistently wall right after you q, so you temporarily displace them and force them into the q.
All in all, a well played Anivia can snowball games, turtle and turn around what seem to be losses, and do a ton of work. I've had 2 games recently where the whole team was not doing well, but I stalled with waveclear, waited for the enemy team to make a bad engages, and gone on to win the game. She is definitely worth learning!