In addition to the above suggestions: Clifton Village, because it is relaxed; The Underfall Yard, a boat building yard and museum next to the Nova Scotia (I have fond memories of the Scotia from years ago); Broadmead shopping centre if you want to see how the planners of the 1940s & 50s can mess up a city.
The two bridges over the Severn are a bit difficult to get to by public transport. The new bridge is best seen from Severn Beach which is at the end of a suburban rail line from Temple Meads. The journey will show you a side of Bristol that the tourists never see and there is nothing at Severn Beach other than the bridge. The old bridge can be crossed using the Chepstow bus. Getting to the actual bridge is another matter.
Unfortunately the new aircraft museum on the north side of Filton airfield is not opening until later this year.
Bristol is a great vibrant city. The traffic is a bit dire but the city buses are pretty good and most of the city is walkable. Unlike Bath it is not a tourist trap.
JA
Edited By JA on 10/05/2017 19:11:36