I've flown on KLM via Amsterdam.
We were in Copenhagen for several days before a Baltic cruise. We had elementary aged kids, so much of what we did was kid centric and not high end (seems like every recommendation on this board is very high end stuff).
If you like zoos and/or have never seen panda, then go to the zoo since there are only a few zoos outside of China with pandas.
Shopping on Stroget is fun. The LEGO store has some things you can't get outside of Denmark.
Climb the Round Tower, see the changing of the guard, visit the ancient ruins under Christiansborg Palace.
We took a harbor cruise to get close to the Little Mermaid statue without fighting through the crowd of thousands on the shore.
Get a burger at the voted best burger in the world, Gasoline Grill.
We had barbecue and beer at Warpigs, a Texas barbecue joint/brewery.
We ate lunch and had beers on Nyhavn.
We didn't make it over to Freetown Christiania, and not sure I would with kids, but you should probably go over there.
We were in Stockholm for only half a day. We went around old town/Gamla Stan, visited the Vasa Museum on the museum Island, had a couple drinks at the Ice Bar (very cool and very cold). Had pastry for breakfast, meatballs for lunch. Bought a little red horse.
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