I think everyone has taken one of the harbour and canal cruises from Nyhavn when they first come to Copenhagen and it is a great way to learn about the history and geography of the city but it is also quite expensive.

If you just want to experience a ride on a boat in the Harbour and see most of the main sights (but not the Mermaid) for the price of a public transport ticket you can ride the 991 or 992 Harbour Bus from one end of the harbour to the other. You can even go out on deck and take photos or simply enjoy the sea air.

The little yellow timetables on board show a map of the routes with the key buildings marked. A great way to enjoy this beautiful city.