First of all it's really good that you're thinking about the team composition and rounding it up, it's good to see people who don't think only for themselves XD
About mages - I would say you've had sam bad luck in your choices. Anivia is one the difficult champions due to her skillshot dependance. A good Anivia wins games, but don't feel ashamed to leave her aside for now, it took me a long time before I could do well with her and even now she's not my forte XD
Cass is very close to her in difficulty. Riot wanted to make a very intensive caster to challenge some of the hardcore players. You've picked among the hardest
If you don't feel comfortable with Annie, I suggest you try Veigar, Nunu and Fiddlesticks. All of them are cheap (IP wise, don't worry - you'll get to spend your RP), they're fun and easy to pick up.
Veigar is one of my personal favourite mages, with insane bursting potential and good utility.
Nunu is not a pure mage, but he can be build like one and in all cases has excellent magic damage. A well executed ult feels very satisfying.
Fiddle is the incarnation of the AoE in mages, his ultimate can destroy whole teams and win battles. He's not too easy but I wouldn't call him hard. Just takes a bit of practice. He, as well as Nunu, are able to jungle, so feel free to try that.
If you want to spend your RP on a challenging mage I will recommend you consider Karthus. He's specific and a bit tricky, but also very enjoyable and rewarding.