Copenhagen is something like my second hometown. I don't live there, but frequent it so often that I know the city by heart. Because I used to live in the south of Sweden, Copenhagen was the closest main city I could travel from internationally, either by train or plane at that time.
Copenhagen is a city that, like Amsterdam is really bike friendly. I can highly recommend getting a rental and discovering the wealth of parks and area's around. Every time I visited, I met or was with someone that was new to the city which gave me the opportunity to discover it with a fresh pair of eyes once again.
If there is one thing I have to suggest you do; is to grab lunch at Atelier September, choose the advocado sandwhich and be surprised. I ended up making countless of these at home, but they never came close to it. The bread is so good!