I'm still a beginner at LoL, but would really like to master Anivia.
Ok, Anivia is a nuker that takes a LOT of finesse to play, since she is the squishiest character in the game, and she's based on skillshots. I've seen a lot of good Anivia's, but also some really terrible ones. Don't play her until you've got experience with other mages.
Try starting out with the similar Ryze and Annie, who can still perform well as magical carries. If you like them, then you can try Anivia once you understand generally how nukers work. Also, Anivia is more support-oriented than other mages, so you do need a good team to get her to work well.
I learned that with Amumu, Morgana, and Kassadin: they don't work well unless you have a reliable team. Anivia is the same way.