I was in Copenhagen for work last March. On a Monday, my partner and I just walked around the city to check it out-- there's a lot to see in walking distance! From the Radhus (city hall-- it's open to the public so you can go inside the building and to the inner courtyard), down the Stroget to the Rundetaarn (Round Tower- an observation point to look out over the city- small fee), to the section with all the museums and government buildings (didn't pay to go in anything formal, just walked around looking at the awesome architecture from the outside and sometimes going inside the public buildings to check them out), ending in Christiania (the free city within a city-- an interesting bit of Copenhagen history and current alternative society). I really liked Christiania- it felt welcoming to me as a tourist even though it's a real neighborhood with people just living their lives. You can get a drink and a snack and hang out a bit, probably find someone to talk to if you want to, or just people-watch. And dog-watch: dogs (and cats) just roam free so sometimes there's some interesting dog pack drama around food or turf or something; like a bit of reality tv/animal planet.
A dinner recommendation: Tight. A place to get a really good burger and a beer, or a fancier entree. It's a little pricey, but what isn't in Copenhagen? I loved the "hyggelig" atmosphere and friendly servers who like chatting with visitors... and even the owner, a young guy, might be there hanging out and visiting with diners. He stopped at our table for a good 15 minutes and we talked about America and Canada and random stuff.