Harborne and Moseley both have really good charity shops, and are a fairly short trip on the bus from central Birmingham. If you're willing to travel a bit further out, Leamington Spa has very good charity shops and is around a forty-minute train journey from Birmingham. You might have luck finding a bicycle on Facebook. I see a lot of people selling them on a group called Fab n Fresh, which is aimed mostly at UoB students. There's also a place called The ReUsers in Sutton Coldfield (which is a twenty-minute train ride from New Street station) that sells second-hand bikes, although quality and pricing can be a bit hit or miss and you may well be able to find a shop closer to central Birmingham.