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!

I don't think it is in the way Bernie Sanders suggests.  What works in Denmrk wouldn't necessarily work here, and not everything of the kind Bernie Sanders talks about works in Denmark.

Summer, although it can be fun to go at other times as well.  St. John's Eve ("Sankt Hans") is June 23.  That night there are bonfires all across Denmark, with witch effigies on top.  The fire sends the witch flying off to Bloksbjerg in Germany.  People gather around the bonfires to sing and celebrate.  July and August are warmer and sometimes drier.  Christmas is fun too.

Top three attractions that I personally would recommend are the National Museum of Denmark, Kronborg Castle in Elsinore (the setting for Hamlet), and boat tours of Copenhagen Harbor (which is clean enough to swim in).  Most people also visit Tivoli Gardens, the amusement park that gave Walt Disney the idea for Disneyland, and there are numerous other great places to go.

Hi Ray.  I can't claim extensive experience with Danish schools, but from what I know I can think of two things.  First, the Danish school year is longer (and summer vacation is correspondingly shorter).  School ends in late June and resumes in early August.  Obviously that wouldn't be very popular with students here, but I think that there is research indicating that a longer school year has benefits in that, when the summer break is not so long, students are less likely to lose momentum and forget what they've learned.  Second, my unscientific personal impression is that the Danish system does a better job of providing vocational training and of respecting those who choose that path.  There appears to me to be more acceptance in Denmark of the idea that not everybody has to go to college.  And with that acceptance comes more respect for people who do not have a university education, but are nevertheless skilled, such as artisans and craftsmen.  Thanks for your question!  Doug