If you want to see the Copenhagen Harbour you can either take one of the sightseeing boats that will take you into the canals and the main harbour, while giving you a running commentary about what you're seeing. Or you can take the cheap way by using the water buses that run between Teglholmen in the south and Refshaleøen in the north with quite a few stops during the trip. And you'll get the whole trip for a single bus fare. You'll have to make your own commentary, though.
The video here is a time lapse I did the other day while taking the whole trip, and then some. And to the other passengers; sorry if I hugged the railing, but I didn't want to risk loosing my camera overboard.