I live here and recommend it for a mini break or much longer. The shops, by the way, are now open in the city until 5pm Saturday and open 10am Monday. However, during December they open each Sunday too.
Take a canal boat ride to cover many buildings and monuments quickly. Everyone is amazed at how tiny the famous Little Mermaid is.
Rosenberg Castle near the centre of town is open for tours and you can see the crown jewels. Nearby is the Round Tower, which gives a fantastic view across the city.
Avoid Tivoli in favour of the Bakken fair (30 minutes from the centre on a train). At four hundred odd years old, it is Europe's oldest funfair and has a carnival like charm that beats Tivoli's middle class appeal in my opinion. The scary ride mentioned earlier is there.
The Carlsberg factory North of the centre does free tours with two free beers at the end, or soft drinks for children. Nearby is the Experimentarium - a great afternoon of hands on experiments for children up to about fourteen and fantastic if the weather is cold and wet.
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