I moved back to Bristol (having grown up here) after living in lots of big cities (Tokyo, London, Sydney) and although it obviously feels small in comparison, there is an awful lot going on, so it doesn't feel quiet or parochial.
The choice of area really depends mainly on your budget for housing. Bishopston/St Andrews is a very good, family-friendly but still central area. Houses in St Andrews are mostly late Victorian, gardens quite small, the Bishopston side has Victorian plus some 30s and 50s houses. I think you could get a really nice family-sized house in this area for around £400k, give or take £100k depending on precise location and size. Lovely park, good shops, lots going on, schools very oversubscribed - but unfortunately that is true of nearly all the family-friendly areas in Bristol.
Redland and Cotham are similar to St Andrews but with bigger houses, a lot of which are now flats or shared houses (lots of students as it is close to the university), hence the parking problems mentioned, but that is improving as residents' parking zones are rolled out. Couple of good primaries, two good secondary schools (but that's probably looking a bit too far ahead for you). Big Victorian semis (5 bedrooms or more) round here tend to be about £600k but you can get smaller ones or ones needing work for less.
If you want somewhere a bit quieter but still with lots of amenities, the Westbury Park/Henleaze area might suit: a couple of good primary schools, high street with lots of independent shops plus Waitrose and a little cinema, close to the Downs. Housing a mixture of Victorian, 30s-50s and some more modern. Gardens a bit bigger on the whole than St Andrews.
Montpelier is very urban, lively, cheaper, rather less family-oriented. Totterdown and Southville are rapidly gentrifying, increasingly popular with families; housing is mostly smaller Victorian terraces.
I think really you need to come and spend a few days having a wander round and getting the feel of places.