In Copenhagen, new neighborhoods are being established, which I think are exciting to compare with the old neighborhoods to be inspired by architecture, environment and opportunities.
My suggestion is to start the cycle tour in Carlsberg Byen, which is located on the site where the Carlsberg brewery was located. You can then cycle through part of Vesterbro down to Fisketorvet and over the bridge to Islands Brygge, which was an industrial port, but which today is primarily a recreational area with cafes and eateries.
When you cycle along the water, you come to Christianshavn, an old district from the 17th century.
If you continue cycling along the water, you will reach Refshaleøen. On Refshaleøen there are new residential areas, restaurants and concert venues.
Finally, you can cycle to the Opera, which opened in 2005, and take the Harbour Bus to Nyhavn.
Have a nice trip!