Horfield, Ashley Down are inner suburbs with nice period properties and lots of great independent shops and cafes on Gloucester Road.
If you'd take location and compromise on a smaller property then can be more central like Cotham, Montpellier, Hotwells, Clifton, Redland....but much pricier.
Bristol is an inside out city - the grimmest parts are on the edges, not in the middle (Southmead, Knowle West etc). Post war estates built on the edges after the war with very poor connection into town.
Traffic is dreadful. The bus service is also rubbish compared to London.
There is a tiny train line that runs from Temple meads (the main station) and wends it's way through Montpelier, Redland and Clifton which would get you there without having to be in a traffic jam.
If you need to get to London for work regularly, take the train from Temple Meads or if you look at Horfield, you could drive out to Parkway and get the mainline from there.
It's not a great city for cycling but they are starting to make it better and it is the quickest way to get around if you are confident. Some steep hills in some parts though so strong thighs needed or an ebike might be worth a look!