Anivia has an incredibly unique playstyle. If you're not willing to put in 30 games to get used to it and get better, don't waste your time.
That said, she's extremely fun, super high impact, and incredibly safe. She doesn't have any hard counter mid matchups, although a good Janna support or Cho'gath top are INCREDIBLY irritating to play against. She is very effective as a counter-assassin. Once you learn your matchups, you can turn into quite a bully in lane post 6 with some kill potential even early. You receive ganks very well and are not horrible as a roamer if you have lane control.
Mana. You need it. These days, don't fool around with morello or grail, they don't give you even close to enough. Go for archangels+rod of ages. Start faerie charm 4 hp 2 mana sight ward, try to get your catalyst on 1st base. Then get tear, finish your rod of ages and go from there. You won't get as much early AP as your lane opponent; don't let it bother you too much. You bases are higher than pretty much anyone out there. As long as you have mana, you'll do as much damage or more than nearly anyone out there.
Based on the champs you say you play, I'd be very surprised if you like anivia, but I could be wrong.