The Queen's residence.
I wouldn't spend time going to the museum, but see the castle from outside and maybe watch the changing of guards. It's a fun, very oldfashioned, short ceremony taking place every day at noon.
A little pricey, but definitely worth it - Especially on a nice day.
Some of the boats will have guides speaking English.
This is pretty much mandatory when you go to Copenhagen.
I would recommend, you put on some comfortable shoes and walk from Nyhavn to Amalienborg and through Kastellet to Langelinie where The Little Mermaid is located. It's a beautiful, very scenic walk.
Tivoli is an amusement park in Copenhagen. It was founded in 1843. It is a very charming and quirky (some might say tacky) place, with old as well as new rides and a wonderful garden.
The Castle where Shakespeare's Hamlet is set.
You will have to take a train from Copenhagen to Kronborg (about 45 minutes from downtown Copenhagen). This website might help you find the right one: www.rejseplanen.dk (It's also available in English).
The old harbor area of Copenhagen. The cafés are tourist traps, but get some refreshments at a grocery store or kiosk and go sit down on the port edge with the locals. The atmosphere there is great.
The freetown. They have their own rules and refuse to follow common Danish law.
It's a quirky little town in the city, with obvious marijuana shops on the streets and very friendly citizens. Although it might seem like a sketchy place, it is completely safe, and it's definitely a interesting experience. Make sure not to take photos or your camera will be confiscated.
Smørrebrød: The famous Danish open faced sandwich on rye bread.
Probably the most typical Danish dish there is.
"Slotskælderen" is within walking distance from Nyhavn and offers a very classical smørrebrød menu.
Smørrebrød is traditionally served with Danish pilsner and snaps.
A very famous conditori. They have the best cakes ever. I'd recommend the "Sportskage". Don't let the sporty name deceive you. Lots of cream and calories and awesomeness!