Depends on your budget really. Long Ashton is nice but pricey, close enough to the city to easily commute, close to most MTB trails and the Weston Road that runs through it is very well used by roadies heading out towards the Mendips, though it has a ridiculous number of potholes.
If you’ve got a shed load of money you could go somewhere a little further afield like Chew Magna, Wrington, Langford. Pretty, chocolate box villages with 4×4 driving torys and old people living there.
Southville is the city option. Very close to the centre but really still really quiet and close to the MTB and good road riding also.
On the not quite so expensive but a bit bland family friendly front, there’s Keynsham and Saltford (good schools if you have kids), which are commuter towns to the east about half way to Bath. They’re pretty dull though and if you want a few beers in Bristol you’ll get fleeced on the bus or worse a taxi. Good road riding from there though, loads of quiet, winding, undulating roads.
There are a few places to avoid in Bristol: Southmead, Knowle West, Hartcliffe, Henbury, Lawrence Weston, Lawrence Hill (not an awful place, just designed and laid out by a moron with a total disregard for taste and common sense). More affordable but still quite nice areas are Redfield, bits of St George, Easton, bits of Whitehall, bits of Bedminster, Knowle, Totterdown. I don’t like North Bristol so I’m ignoring that all together.
If I could chose and afford it, I’d live in Southville or Long Ashton. I actually live in Redfield, which is nice in a slightly shabby, up and coming, very diverse kinda way but a bit of a pain for road and mtb.