I think a lot of people would say the best thing to do in Belize City is to get out as quickly as possible.
In fact, there are some interesting sights in the city. The main point of interest is probably the architecture -- the old colonial buildings, especially in the Fort George area, and the ramshackle houses and stores. The city has a wonderful Graham Greene-like ambiance, if you have a taste for that sort of thing. Beyond that there is St. John's Cathedral, the museums at the Marine Terminal, Government House, the Bliss Institute and the Lighthouse.
As for shopping, there's not much. The Image Factory is the best gallery. There's the post office for stamps and the crafts store. Albert Street is the main commercial street.
You can gamble at the Princess Hotel (but don't stay there) and get a good meal at a number of places including Four Fort Street, tghe Smokey Mermaid, Macy's, Three Amigos, Searock and others. Except in the roughest areas, you are okay by day, but after dark take a taxi (it's safe enough to walk short distances in the Fort George area.)
Belize First Magazine http://www.turq.com/belizefirst/