Welcome to Brum! I would say that posters above have made some good suggestions, but I wouldn't necessarily consider some of them to be "birmingham"... you may be quite surprised at how long it would take to get to those southern villages into the city centre
I don't think Moseley is a dump at all! And it has some fantastic restaurants, a great market etc... it depends what you're looking for. I've only visited there so I don't know much about it but I've always thought it was lovely.
Personally I would look at Bournville. You could get something amazing in Bournville with your budget. The primary and junior school is rated outstanding and the area has a lovely village feel. The only thing that's missing from your requirements is the high street but you have got some small shops and amenities there, plus you've got Cotteridge and Stirchley high streets (- Stirchley high street is really up and coming for little independent places to eat and drink. Do you like craft beer? If so you really should go there for a drink and some lunch after looking at houses. It's about 2 mins walk from Bournville train station).
And if you jump in the car, (or 1 stop on the train, once they've finished all the integration building work around Selly oak station) then you've got Selly oak retail park just a few mins away. Massive new Sainsbury's, M&S food, Next, Homebase etc etc. Whatever you need.
Plus Bournville is only 12 mins to the city centre on the train, or maybe 20 mins drive? You've also got the QE hospital right there in Selly oak, as well as the uni.
I've heard great things about Sutton Coldfield and Solihull, as well as so many other little southern villages like Knowle etc, but to me they seem so far away from the city! And, (having come from a place north of bham myself, so I'm allowed to say this), I would say that the areas to the south of bham seem a bit nicer in general, eg if you wanted to pop to Worcester for a day or whatever.