Edgbaston is very nice and close to city centre. Don't know about fees though I would think not too cheap. Ask Johnny Giles he is a member there!
Moseley is also a nice course, again not too far out of city centre.
A bit further south you have gay hill, kings norton, ladbrook park. All lovely courses, gay hill used to have good social too.
I could go on as there are lots of options but if I were you I would pay and play quite a few to begin with then go for your own preference.