There are presently 20 active breweries in Bristol, some of which have their own tap rooms... New Bristol Brewery; Bristol Beer Factory; Left Handed Giant Brewpub; Little Martha; Moor Beer plus a couple already mentioned. The Shakespeare on Victoria Street (not the Shakespeare mentioned earlier!) is the tap room for Good Chemistry.
The Cornubia and the Seven Stars (both mentioned earlier) are great cask ale pubs, though the Seven Stars could do with an internal facelift.
I'd recommend the Gryphon on Colston Street for dark ales, though you have to like heavy metal music! The Strawberry Thief on Broad St is a Belgian beer bar. The Bridge Inn on Passage St, the Barley Mow on Barton Road.
On King St, you'll find the Llandoger Trow which is an architectural treasure; and the Old Duke if you like jazz. There's also the Beer Emporium, which occupies three former wine cellars underground - mostly craft keg beers with occasional real/cask ales.
A couple of favourites of mine for cask ales are The Volunteer Tavern, New Street; Swan with Two Necks, Little Ann Street; and the Lime Kiln, St Georges Road.
All these and more (many, many more) can be found at
WhatPub.com, which - being a CAMRA (Campaign For Real Ale) website/app - focusses, but not exclusively, on real ale pubs. In theory, WhatPub lists every single pub in the UK so is a useful guide.
The Bristol branch of CAMRA is very active and there may well be opportunity for you to contact them and join with some of their activities. Facebook page is CAMRA Bristol Chat.