This is by no means comprehensive, but are where I’d go:
Noma. It’s no El Bulli, but this is open, so! I love how it looks like a nondescript warehouse from the outside.
The Opera House. Ridiculous inside. (It looks like a pumpkin.) The Royal Danish Ballet and Royal Danish Orchestra both perform here all the time and apparently THE QUEEN comes all the time.
- http://media-cdn.tripadvisor.com/media/photo-s/01/b0/e6/4f/the-royal-danish-opera.jpg
- http://media-cdn.tripadvisor.com/media/photo-s/05/64/15/bb/the-royal-danish-opera.jpg
Koncerthuset. It’s a theater with 4 semi-transparent screens for walls. So you can see inside during the day and watch projections on it at night.
The Museum of Danish Resistance. Apparently this place BURNED DOWN but everything inside is safe, and you can go see it in a temp location.
Nyhavn. The waterfront districts, obviously.
Tisvildeleje. Seaside village about an hour away from the city.
And lastly, GO BIKING. It is unreal coming from NYC.