Here's a sampling:
• Tivoli Gardens is by the main train station. It's an urban theme park said to have inspired Walt Disney to make Disneyworld.
• Copenhagen is a foodie city. Some of the top-rated restaurants in the world are there like Noma, Alchemist, and Geranium if you can find reservations. It's also a town known for great food halls as it was one of the first cities to invest heavily into them. Be sure to try some smørbrød, which is kind of like rye bread topped with unique toppings, just like you'd fill sushi.
• Although a fraction of the size of the Amsterdam ones, Copenhagen has some canals and there are plenty of tours to get a ride around them and see sights on the way.
• Freetown Christiania is interesting. It's an anarchist commune in old military barracks that doesn't consider itself part of Europe. It has a lot of unique art, crafted goods, homestyle comfort food, and graffiti, but a lot of folks went for the open air drug market. Don't take photos anywhere where there's people in them.