Being a girl my idea of entertainment in Birmingham is probably very different to your 13 year old boy's!
For shopping there's The Bull Ring which leads out into Birmingham centre, that's pretty huge although not sure it'd be massively entertaining for a boy...
Cadbury World is a few miles away from the Bull Ring and New Street train station...
If you have a car you could drive down to Kidderminster to the safari park - I'm 21 and still love the place! The animals are used to the cars now so interact with the public a lot, and it's pretty cheap to get in...