Depends what you want - areas are all fairly defined.
Clifton - lovely area, expensive, lots of students in term time, nightmare for parking!
Montpelier - 'bohemian' feel - if I wanted to live in the centre, I would live here, as is little oasis. However, quite close to St Pauls/Stokes Croft which isn't so great. But easy access to trains (has its own station on branch line). Close to Gloucester Road for shopping (lots of indie shops, not so any chains), and also centre (including new Cabot Circus - which is nice).
Redland - leafy, green suburbia. Has train station for link onto branch line, but I've never felt is has much 'community spirit' there like some other areas have.
Henleaze - familyish area - nice, local shops, rubbish public transport links.
Waterfront - lots of 'glass flats' - lots going on, festivals etc, lots of pubs down there. Not beautiful houses though.
Sneyd Park/Stoke Bishop - the 'posh end' of Bristol. Big houses, moderate gardens, trees.
If schools really aren't an issue I would go for Montpelier tbh. I love it down there, but schools would mean it wasn't an option for us atm.