The train runs directly from terminal 3, platform 2 to Copenhagen. From Copenhagen Central Station, it is just a 2-minute walk.
If you take a taxi from the airport, it will cost around DKK 250. All taxis run on a meter and accept dankort and most major international credit cards.
Take a train from Østerport station and get off at the central station or take bus no. 26 heading for Ålholm Plads and get off at the central station.