I would recommend a trip to Frilandsmuseet in Lyngby (open air museum). for a further trip, try the area around Hillerød-Fredensborg. The king Christian 4 build some nice castles there, and they still try to maintain the parks to a renaissance standard. Also a large forrest with wildlife and exercise-enthusiasts are at hand there.
Brede near Lyngby has a part of the National museum as well as a collection of crafts in Hjortekær (a gate to Dyrehaven).

The area of Frihavnen (near Østerbro) provides a lot of harbour activity, now more like sailboats.
Roskilde has the Viking museum and the Roskilde Domkirke (Cathedral) , where most of the danish kings and queens are buried.
To the west of Roskilde, which is a relative small town, there is a lot of farmland, but follow the shore to the north, and you find a beautiful nature.
A bus travels from the station. heading Skt. Jørgensbjerg / Herslev. You find a very good brewery in Herslev, a small former fishing harbour. I think they are open to visitors and taste on fridays.
I can also recommend the GB-beer you can buy in Roskilde, but I'm not sure if the brewery can be visited.