Take a cab to the Radhuspladsen (sp?) (Town Hall Square), visit Tivoli on one side (next to the Carlsberg museum also) and take a stroll down the Stroget (pedestrian shopping streets) on the opposite side.
It's a great city, but expensive to eat and drink. One of the best steaks in Europe is from the Hereford Beefstew backing on to Tivoli, which also has a micro-brewery next to it. Also try the marinated salmon (Gravilax) starter, melts in the mouth! Hmmm, I'm drooling...
For Euro 2000, they had giant screens in the square, with teriffic atmosphere too.
Nyhavn also recommended, but bit of a tourist trap and more expensive dining...
Enjoy - SLF