If you only have three days, you'll probably spend most of your time in Copenhagen. Come back again for more time so you can see the rest of Denmark!
In and near Copenhagen, in no particular order:
Tivoli Gardens -- Every Copenhagen visitor goes there, so you will. The evening fireworks in the summer are nice. https://www.tivoli.dk/en/. If you like amusement parks, there's also Bakken, the oldest amusement park in the world. It's about 10 minutes from Copenhagen.
https://www.bakken.dk/english/. From Tivoli go over to Strøget, the walking street, for shopping. https://www.visitcopenhagen.com/copenhagen/stroget-0.
Kronborg Castle -- Kronborg, about an hour north of Copenhagen on the train, is where Hamlet is set. In the summer, scenes from the play are enacted all around you. http://kongeligeslotte.dk/en/palaces-and-gardens/kronborg-castle.html. On the way, stop at Louisiana Modern Art Museum. https://en.louisiana.dk.
Christiania -- This hippie-run "free state" in Copenhagen has a "green light zone" where they sell cannabis. It's illegal, but the occasional busts seem to be aimed mostly at the dealers, which is why there are these rules: have fun; don’t run (might be a raid!); no photos (might be evidence!). I hear you may now be able to take photos, but it's probably good to be discreet. https://www.visitcopenhagen.com/copenhagen/culture/alternative-christiania.
Vesterbro -- One of many cool areas in Copenhagen, where I've found great Airbnb rentals. https://www.visitcopenhagen.com/copenhagen/hipster-vesterbro.
Copenhagen Harbor -- Bring your bathing suit, because Copenhagen Harbor is so clean you can swim in it. I also enjoy the boat tours leaving from Nyhavn. https://www.visitcopenhagen.com/copenhagen/sightseeing/boat-tours.
National Museum of Denmark -- Here you find great artifacts from Denmark's long history. And a good gift shop. https://en.natmus.dk.
New Carlsberg Glyptotek -- Here you find an amazing collection classical sculptures. I've seen more here than in Athens. https://www.glyptoteket.com.
Carlsberg Brewery -- Speaking of Carlsberg, you can also visit the Carlsberg brewery. http://www.visitcarlsberg.com.
There are many fine restaurants with Michelin stars in Copenhagen. Reserve well in advance. If you can't get into Noma (http://noma.dk), try Geranium (http://geranium.dk).
The famous Little Mermaid. She's little all right. You can see her from one of those boat tours leaving Nyhavn. https://www.visitcopenhagen.com/copenhagen/little-mermaid-gdk586951.
And much, much more when you come back for a longer visit.
I'm happy to answer questions!