If you want to go to central stations like Kongens Nytorv or Nørreport, you should get on the metro.
Read more about the metro here.
If you want to go to Copenhagen Central Station, the train is your best option.
Read more about the train here.
Bus 5A will take you directly to Copenhagen Central Station, City Hall Square, Nørreport and other stations. It takes about 30-35 minutes from the airport to the Central Station.
Read more about the bus here.
Taxis can pick up passengers outside terminal 1 and 3 and will get you to downtown Copenhagen in about 20 minutes depending on traffic. It will cost approximately DKK 250-300.
Copenhagen Card gives you unlimited public transportation in the entire Capital Region of Denmark incl. public transportation to/from Copenhagen Airport.
Read more about the Copenhagen Card here.
With a City Pass you can enjoy unlimited travel on buses, trains and the metro in Copenhagen in zones 1-4, from the moment you arrive in Copenhagen via the airport.
Read more about the Citypass here.