Raising a kitten?

I've owned dogs, rats, rabbits, etc. but recently I decided to adopt a kitten and I realize that kittens are hyper active and attack everything because they are practicing their instincts. However, even if I just calmly reach over to pet her she wraps herself around my arm and bites down. It's not hard enough to draw blood, but it still hurts. (She's only bitten hard enough to draw blood once and it was because I was holding food). So, how do I assure that this is just a phase and she'll be nicer as she matures? Also does getting her spayed affect her personality? Because I wasn't going to spay her since she's an indoor cat, however; if it'll make her calm down a bit then I would.