we very nearly moved near bath 18 months ago...we would have been there now if it wasn't for a family illness...all sorted now and perhaps we're going to look into it again.
my sister lives in bath and we just fancied it...really nice city, googd for cultural/intellectual things as you say, but lovely countryside etc nearby and the west country proper on your door step...
bath proper is quite expensive, and city/suburban living there would be just as it is in any other city i suppose, so we were looking outside in the semi rural villages for a nice new family way of life.
we looked at villages to the east ...bathford, batheaston are just outside the city. proper countryside a bit further east is nice...box (funny name!)..and around are nice and only a few miles from the city.
we looked further south towards frome too...am sure there are some nice places round there...
not really sure about west of bath..towards bristol didn't appeal really, but again, would be good to get there for culture stuff.
we're still looking (half heartedly - thinking about moving to the coast now!), and have noticed slashed prices on rightmove from march when we last viewed a house down there (in atworth...again, quite nice)
keep us informed..might well end up there one day ourselves