Is most of Bristol doable on foot? Any specific recommendations on what to see?
Depends on how fit you are and what you want to see.
The Whole of Bristol isn't doable on foot as it's 110km but I'm guessing you just want to see the city centre?
Give some idea of what sort of thing you would like to visit and then a plan would be easier. You can do the docks and it's attractions, then the cathedral, the museum/art gallery, take a bus up to Cliffton and explore the attractions then walk down. That would be a busy day
the central area is built around the floating harbour and therefore is flat and easily walk-able. However the rest of the city (a bit like Rome) is built on the hills surrounding the centre and much harder going. Your best bet is to start at the free M Shed where you will get a brilliant overview of Bristol. From there a ferry boat ride will give you a different perspective. I recommend the oldest part of the City around Corn Street, St Nicks covered market at lunch time as there are some amazing food options for people on the go. Castle Park if the sun is shining to eat your picnic. I posted a walking tour guide on the things to do page a while ago that includes all that and some more.
A day rider on the bus will cost £4 which means you don't have to walk up Park St towards Clifton and the Downs.