I think the best way to appreciate the city is to stroll without a precise destination - just take a look around.

The town is well connected by underground, buses and urban trains and it is full of interesting shops and restaurants (like Banksia, a lovely Australian kitchen and beer bar).

I sadly missed the botanic garden and the Rosenborg Castel because I had no time - they must be wonderful.

I absolutely recommend the Paper Island to enjoy a typical street meal; there is food for every taste: Brazilian, Arabian, Italian and much more.

If you want to stay more than three days in Copenhagen I personally suggest this AirBnB solution: next to the metro station, just three stops from the city center, fabulous apartment, super nice hosts.

Thanks Casper&Lars!