South Birmingham
Moseley can be nice, unless you're on the sparkhill border.
Kings heath is nice, unless on the druids heath border.
Transport is buses at the moment, crime not anything I've had a problem (moseley on the sparkhill border is bad though)
(I work in moseley and lived in kings Heath).
Selly Oak and edgbaston have some nice houses. In selly oak it's best to be on the bournville border and edgbaston near the harbourne area (otherwise you're in student areas).
Harbourne and bournville are nice. But harbourne is terrible for traffic and Bournville isn't as nice near kings norton.
Good transport links (bus and train) in all these areas, again I've not had any crime problems.
(I've lived in selly oak on the bournville border, worked in harbourne and studied in edgbaston).
Solihull is nice, hall green is okay. Shirley is where I plan on buying a house, on either the hall green or solihull border. I live near Shirley now, but nearer to yardley Wood. I'd avoid the yardley Wood area to buy.
North Birmingham
Sutton Coldfield is your best bet. Steeetly is okay, not much going on round there though. Transport is better is Sutton as there's trains. But stay away from anywhere near the kingstanding borders. I lived in kingstanding and it's awful, loads of crime there lately, stay far away from there.
As for a little outside Birmingham, houses in Tamworth and Knowle are nice, I've got family in both. But houses in Knowledge tend to be quite pricey. Tamworth's not too bad and there's decent transport and shops.