Being the typical gastrotourist you will probably not cover much more than Copenhagen, allthough you should also have a look at Jylland. If you go by train or car to Århus – or venture to Henne Kirkeby Kro in western Jylland, you are not far away from the town of Vejle (and by the way, close to the terrific gourmet restaurant Ti trin ned in the town of Fredericia).
In Vejle you should stop for lunch at restaurant Remouladen, a new place that serves some of Jyllands best smørrebrød (open face sandwiches). This truly traditional danish specialty has for many years been neglected in a gastronomic sense outside of Copenhagen, but now it is spreading again.
Remouladen is proof of this and the place is very pleasant, modern and clean. It is part of a group owning a good microbrewery, so you can have a good beer as well.