If you want a short-ish commute I’d suggest SW Brum. I live by the Lickey Hills and ride to work in the centre – about 30 mins via road/canal. I have a reasonable amount of local riding on the doorstep and the M5 and M42 within 5 minutes drive. You can make a long ride out to the Clent/Wyre Forest area too.
Alvechurch and Barnt Green have a village-y feel and are <30mins by train on the cross city line to the University.
If you want ‘cool’ try the Moseley-Kings Heath or Harborne areas for the young-uns. Too built up for me though.
Alternatively, on the north side of the city centre, Sutton Coldfield or SE, Solihull area.