The museum is excellent, the Cathedral is cathedral-y, I recommend the Fat Pig pub for both food and boozes and The Flat for the best pizza you will ever have, which just happens to be vegetarian. There's a halfway house in the area mind, so quite a lot of street-attached folk hanging around, no point pretending there aren't. Just be sensible.
If you've got £££ then Rendezvous does
excellent food, just don't look at their website, don't think it's been updated since it was created in the 90's
The Phoenix Art Centre is great and has loads of bands on as well as art exhibitions and a nice cafe bar. The Cavern is an outstanding music venue, loads of bands and themed nights, although the latter are very studenty. Both in Gandy Street
The Quay is good, Topsham Brewery is a good booze barn for all your modern craft beers, they'll also have at least one old man beer if that's your thing and a pizza van outside that opens in the afternoon/evening, they also have films and music upstairs. Or there's the Prospect for a traditional pub vibe and reasonably priced pub grub. There's also loads of I-saw-you-coming little shops under the arches that are worth a mooch and a huge bric-a-brac shop. Plus you'll pass by/over a section of the Roman wall on your way down.