There are many nice area of Bath to live and have friends who live all over Bath and love the many different aspects of living in a city centre/outskirts and beyong.
However I don't have a great amount of advice about house prices and schools as I live in a surrounding village.
This thread and this other thread are on the talk boards, though they are originally quite old, some of the information about the schools is still quite current.
Bathampton is a very popular village community just on the outskirts of Bath, I've heard the schools are great there. Prices are a little more reasonable, with some ex-Local Authority houses with good sized gardens coming up now and then, you do get more for your money than you would closer to town. Though you can walk into town, they are a good 35-45 minute walk, sometimes along busy roads. Bus service is good.
Widcombe and Bathwick are within walking distance of the centre, but are more pricey. They also have good schools.
Bear Flat is nice, a downhill walk into town, more terraced houses than semi's. It hasn't got the same villagey feel to it, but seems popular with some of my mummy friends. I'm none too sure about schools as my friends are preschool parents.
To the north there is Camden and Lansdown, they have the most amazing views, a downhill walk into town about 20 minutes or so. Prices are a little high as the views are so good.
At the moment Weston is quite desirable among friends, it is a walk into town, again buses are regular. It's not really got a villagey feel, but is quite popular among families and house prices are quite reasonable.
I live in a village outside Midsomer Norton [about a 25 minute drive into Bath] and I love our location. My DD is a preschooler, so though we know where the schools are we aren't yet having to make the decision on where to send her. I love Bath, it's a great city, it's easy to walk about [as long as you don't mind the odd bit of hill], has a great park, some lovely cafe's and has a great atmosphere. If you live in town, it's a short drive to some really great outdoor spaces [dyrham park, Lansdown etc].
At the moment the Mumsnet Local is being revamped, some of the options aren't quite sorted, but it's getting there. Please feel free to ask any questions and hopefully others will be along soon to give some advice as well!
K.