It's surely a great experience to see a new country and go around its capital, i enjoyed that fact. Well, to tell the truth, Copenhagen has a lot of nice places and is a really beautiful city in general, all the squares, the old buildings, churches and towers, the small alleys and pedestrian streets are just wonderful.
A must see in Copenhagen is Nyhavn, the small port (in the photo above), where you can find a lot small coloured houses stacked next to each other, and the sight is just an artist's imagination, it's a very good idea to take the cruise from there that takes you through Copenhagen from the sea, you'll see almost all famous sights, includng Langelinje "the little mermaid" which is Copenhagen's most famous site, but it's better to visit her from the land side to see her face.
Another great place is the Round tower, it takes you to a thirty somehting height, on a spiral path almost right to the top of this tower. From where you can see all copenhagen, and that's a really impressive sight, and to give it even more pleasure, take the St. Kannikestraede to reach there and buy some delicious caramellized almonds from the handsome Moroccan guy there.
Also, it's really nice to go along the pedestrian street¸get where you can also do shopping and souvenir searching, i bought myself a Copenhagen T-shirt and mug there. And if you like to party, that area around that place is full of pubs and night clubs where you're guaranteed to have some fun and good time.
Also if you're travelling from Norway or Sweden, make sure you take the bridge connecting between Copenhagen and Malmo in Sweden, that's a sight you shouldn't miss especially on a sunny day. Beautiful!
The down side which make me not like the city was actually the people, the city was too dirty and the place in front the city hall was all trashed up, also on Saturday the streets were full of drunks, everybody's walking with beer or other drink in their hands and drink like crazy, swaying people in the street was a normal sight. That was irritating and sometimes even disgusting.
I don't think i'll visit it again unless i have really nothing else to do.
If you have the time and money, go see it, but if you don't don't be so sorry about it.