I'm a student living in Bath. I forget the name of the place with live jazz, but it's in Green Park, near Sainsburys.
Down one of the side streets, near the Abbey, there's this lovely place called Ben's Cookies. I think, it's a month since I've actually been into the city xD But they do sinfully gooey, delicious, yummy, massive cookies of all kinds.
The Baths are lovely. Bath students get in free, so when a friend came to visit we spent hours going round there. Make sure you get your money's worth - the audio guides are really good too, the bits by Bill Bryson especially.