One of the things I would recommend for anyone who is visiting Copenhagen would be to take a look at the Rundetårn (The Round Tower). It's close to the centre of the city and serves as a great place to get a really nice view of the whole city.
The tower was completed in 1642 and was built to house three important facilities: an astronomical observatory, a church for students and a university library. They are all joined by way of a spiral ramp that winds up the inside of the tower. The official reason for this was to allow horse and cart access to the library to move books around but rumour has it that there used to be races between the King and Noblemen. Evidence of this is a couple of medals in a small gallery in the tower displaying things like "fastest time to the top of the tower". It's good to know the King was spending his time profitably.
The library these days plays host to various exhibitions of art, culture and history. When I visited, this space was used for an exhibition called "Hungry Planet". This was a display of photos of families from all over the world showing their weekly food expenditure. I thought it was really interesting to see comparisons of countries from developing and developed countries and also to see how people from two developed countries, like Denmark and the US, have different food philosophies and the different kinds of food products they spend more money on.