In Copenhagen it only takes less than 15 minutes to go from Copenhagen Airport to the city centre by train or metro. Both metro and train run from terminal 3. You will need a ticket for three zones, which costs DKK 38.
If you want to go to central stations like Kongens Nytorv or Nørreport, you should get on the metro.
Copenhagen Card gives you unlimited public transportation in the entire Capital Region of Denmark incl. public transportation to/from Copenhagen Airport. Additionally, you get free admission to over 80 attractions and museums.