DS has a house in Hawkesley, on the edge of Kings Norton. It had a good reputation when he moved there, then it went downhill a bit but I think it is getting better again.
It's right on the edge of the Birmingham conurbation and he can walk five minutes and be in Truly Rural Worcestershire.
There's a local medical centre with a bus stop outside it which has a half-hourly service direct to the (Queen Elizabeth) hospital. (Next to the university.) The bus goes on via Harborne where there is a Waitrose. In the opposite direction the 48 bus goes to Northfield where there is a large Sainsbury's.
Hawkesley has a small shopping square (with a pharmacy), and a Church primary school. Ark Primary Academy and Ark Kings Academy (secondary) are about ten minutes walk away, although I have no idea as to the quality of the education. King Edward's Schools are about 25 minutes away on the 35 bus route (which also goes to Kings Heath, Moseley, and the city centre. Kings Norton train station (on the cross-city line) is a good 20-minute walk, however, and there is no direct bus - you need to change. Junction 2 on the M42 is about 15 minutes away.
Hawkesley Square is on top of Wast Hill tunnel on the Worcester and Birmingham canal. The tunnel emerges near the Academies and the towing path has been upgraded to be suitable for cycling, with a sealed surface. It's a popular commuting route to the city.
I don't know about house prices now but when DS bought his (4 bed) house it was under your budget.