I live in Bath with two small DC and my MIL lives near Sherborne, which is where my DH went to school. We like it here and it's a lovely place to bring up a family. Would you or DH need to commute anywhere regularly (London by train, or M4 access) as that might guide which side of the city you look. In a nutshell:
North - Lansdown - top of a big hill but has P&R bus into town. Expensive due to easy access out to M4. Couple of good primary schools, also home to Kingswood & Royal
High private schools. No secondary schools. Look at Ensleigh new build development - £650k might just get you a 4 bed townhouse if there are any left. No shops or village centre, but a good pub.
West - Weston & Newbridge - both lovely family areas with good primaries and Oldfield Secondary. Prices in Weston are more reasonable, you'd get a nice 4 bed (but poss 59s/60s) for your budget. Both have a few shops.
East - Larkhall / Fairfield Park. Loads of people go nuts for Larkhall, call it "The Village" - it has a deli, butcher etc but I personally hate it as it's all narrow streets and hills! You'd probably get a Victorian terrace for your budget.
South - Oldfield Park used to be a family area but now largely students. Combe Down is nice, good primaries and Ralph Allen secondary. More of a village feel, and has shops, Drs etc. Mix of housing but you'd get a 4 bed 50/60s there but may need work. Bear Flat is popular and a bit cheaper than Combe Down. Decent primaries and home to Beechen Cliff (boys) secondary which is competitive to get into. Twerton/Whiteway/Southdown are not going to be places for you.
Central - Bathwick and Widcimbe both have good primaries, access to train station and city centre on doorstep. Your budgets won't go far though and I personally would t want to live in either due to all the traffic and pollution, although a clean air zone congestion charge is coming in next year which could help.
Feel free to ask me any more specific questions, or post links to example houses and I'll tell you good/bad points on areas!