The Viking Ship Museum is the only place in the world where you can go sailing in an actual Viking boat (the sailing season is from May 1 to September 30). It's located in the historic city of Roskilde, just 30 minutes west of Copenhagen by train.
The limestone quarry at Faxe is the biggest man-made excavation in Denmark, and contains fossils and chalk that go back 63 million years. Plus, it's only an hour's drive outside of the city centre.
Kronborg is arguably the most famous Danish castle of them all - for which it probably owes a lot to a certain English playwright and the role of the castle in his seminal 'Hamlet' play. More than 400 years later, it's still there in all its glory - and totally worth the 40 minute train ride North of Copenhagen to the old harbour town of Elsinore, where its located.
One of the perks of being a small country with a well-functioning public transport system is that you can get from A to B quite quickly - even if 'B' means another country. This is where Malmö, Copenhagen's 'little brother' and main city of Southern Sweden, comes in. Taking just 30 minutes from Copenhagen by train, the trip does require a passport, as there is a temporary border control in Sweden.
Rågeleje Beach is located on the northern tip of Sealand, and is home to these impossibly retro-looking, candy-striped beach houses. The beach and its surrounding areas are some of the most well-preserved in Denmark.
Apart from serving as the official hunting grounds for the royal family of Denmark, The Deer Park is also a UNESCO World Heritage site. It's located just north of Copenhagen, and is reachable by train in just 15 minutes from the city.