There are very frequent buses out along the A38 / Bristol Rd / Bristol Rd Sth which make Selly Oak, Northfield, Longbridge and in to Rednal or Rubery very easy.
Also, anywhere along the cross City Train line (lots of pupils use the train and the station at the University then walk across campus).
As others have said, Northfield High Street is nothing to write home about so people tend to lump the whole area together as being a bit 'meh' but it varies hugely road by road.
Selly Oak to the North of the Bristol Road, behind Northfield Manor Park is nice, but to the South, and closer to the University be wary of living in the part where all the student accommodation is.
I can't speak for Manchester, but in Birmingham a lot of the areas are really mixed. Some houses in Kings North are lovely. Some are okay, some areas of KN come under some of the worst social housing estates but other areas have some beautiful £million+ houses. So it is difficult to say a whole area is good or bad.
Your budget is tight for a family needing 3 useable bedrooms.
It seems an unusual mix to be wanting to pay private school fees when your budget is what it is.