The one with the names

I mean, depends what you're into, obviously. If you like museums, then Copenhagen is lousy with them.

If you like the slightly morbid, Medicinsk Museion is a medical history museum on the grounds of an old surgical college. Lots of freaky body things preserved in formaldehyde, lots of gruesome history in there. There's also Davids Samling ("David's Collection"), which is one of the largest collections of Islamic art in Europe, which I recommend making time for if that's your kind of thing.

It's also well worth making the walk to Nyhavn, it's that lovely colorful harbor row that you will see in every piece of tourist footage taken of Copenhagen ever.

It's a bit of a tourist trap, granted, and the restaurants down there are overpriced to hell and back, but it's a nice scenic walk and you can finish it off by taking the harbor bus (which is a boat) out to the big modern opera house, or else wander down the waterfront for half a mile and see the royal palace of Amalienborg if you want. The grounds there are open to the public, just so long as you do not wander close to the windows and try to peek in. The guards will yell at you very loudly to get back.

You can always take a wander down Strøget as well, that's the big pedestrian shopping high street that runs from city hall down to Nyhavn. It's got all the same shops you'd see literally anywhere else in the world, mostly, but there's nice fountains and statues and a LEGO store along the way if that tickles your fancy.

I don't particularly recommend the Little Mermaid statue, mostly because that place is absolutely stuffed with tourists 24/7 so you'll never get close to her, and also because it's honestly just a fairly modest bronze statue on a rock and not exactly a mind blowing experience to go see. Well, unless you catch her on a day when she's got her head sawed off. It's not happened for a few years now, but it's an "any minute now!" kind of a deal.

The churches in Copenhagen often have music and live performances going on—I heard a beautiful rendition of Mozart's Requiem in a university chapel a couple of years back, and it's usually worth checking their schedules to see if any of them have something happening you'd like to go check out. It's also worth visiting the churches just to see them, we do have some nice ones if you like architecture.

Marmorkirken ("the Marble Church" or Frederik's Church as it's called in English) is also absolutely worth a visit. It's a single enormous dome church with a beautiful interior, benefits of being built by royalty.

There's also a theme park in the middle of Copenhagen called Tivoli. It's one of the oldest theme parks in Europe and it's usually quite packed, but it's worth a visit even if you're not much into rides and carnival games, since it was built originally as a beautiful pleasure garden and retains a lot of gorgeous landscaping that's worth taking a walk around.

Finally, if you want to experience true Danish cuisine, go find a pølsevogn—a hot dog stand. The Americans will tell you that hot dogs are their thing. They are wrong, ignorant and foolish - they are OUR thing! Go get a hot dog with everything on it and you will know the soul of the Danish people!

... unless you're vegetarian, obviously, in which case don't. Our pork-filled hearts are something you must admire from a slight distance instead.