Lived round there for 7 years, bit of a strange part of the world, some nice bits (Edgebaston, Moseley, Bournville, Harbourne) some cracking people but when it’s rough it’s proper rough.
Clent Hills are O.K but Cannock is better IMHO, there’s some loops in the Lickey Hills if you know where to go, they’re very accessible from South Brum.
If you’re going to commute by bike try and live near the canal, it takes you right into the Uni campus and makes everything much more civilized, Birmingham roads and drivers are some of the worst in the U.K. for outright hostility to cyclists. My blood pressure has decreased notably since I used to ride to work in Brum, I was having a proper squeaky bum moment on at least a weekly basis.