Laka Lono Tiki Bar is pretty good. They make really good speciality drinks and have cool decorations. Brickway Distillery is nice.
Most of the restaurants are really good, so I would just pick one based on budget.
If you're going to be there late one evening, and enjoy dancing, go to The Max. (Yes, it's a "gay bar," but a hell of a lot of fun and packed with all sorts of people).
The Durham Museum is just outside the Old Market, but definitely worth a visit. Kaneko has cool displays, too.
There are some neat shops in the Old Market. Visit the Passageway.