братья is the nominative plural, so you could say:
У меня есть братья = I have [some] brothers
братьев is the genitive plural, and the genitive is used in constructions with «нет»
У меня нет братьев = I have no brothers

I think the simplest way to say "my hobbies are" is:
Мои увлечения: [and then list them].
There's also a word «хобби» in Russian, I think you could substitute it for «увлечения» here. Note that «хобби» is indeclinable though, which means that ALL its forms are the same (whether they're singular or plural).
EDIT: Lampada's answer's probably better on the hobbies question, he's Russian