And if you're in Denmark, you might as well go to Christiana, which is a small hippie community within the center of Copenhagen. They don't pay any taxes, and it' very anarchistic but it's still very beautiful. The nature is amazing.
Speaking of nature, I would go to Skane, which is the northern part of Denmark. Again, the nature is incredible. You go the Baltic Sea and the Northern Sea flowing together, and you can stand with one foot in each ocean.
One more thing regarding nature would be going to the southern part of Denmark where we have something called Sort Sol. It's a natural phenomena where you have millions of birds gathering mostly in the daytime, but also at night, and they fly huge formations.
One more thing definitely should do in Denmark would be going to a handball game. It's a Danish national sport on par with football but it's just not as popular in other countries, but still it's a very interesting sport.
Last but not least, I would recommend going to a place where you can eat smørrebrød, which is the Danish national food. It's basically rye bread with different kinds of toppings.