I liked visiting Roskilde for the cathedral and Helsingor for the castle. The Glypotek is a very nice art museum (and has a tropical greenhouse inside). The Resistance Museum is interesting. Just have fun strolling around in the main area of town, on and off the Stroget, through Nyhavn and Christianshavn.

I would have liked to get to the Louisiana Art Museum and another castle (name escapes me). The town of Dragor, near the airport, also sounds interesting.