I've lived in Cotham, Southville, St Pauls, Montpelier, Easton and Totterdown.
Would also recommend Totterdown for good community feel and a short schlep to work. Windmill Hill area borders Bedme but is not a bad spot.
Southville is also good – as mentioned above good access to waterside / countryside. Avon cycleway in one direction out towards bath. Ashton Court and Leigh Woods in t'other.
Had a great time living in Easton but was a bit edgy, much more so than St Pauls to be honest, which is actually pretty gentrified in places, especially where it edges onto Montpelier around the top end of Ashley Road.
Cotham / Clifton pretty much as described above.. nice areas but full of loads of hoorays and nightmare parking.
Bristol is a great place – you'll have a ball.