When going to Copenhagen City Center, you can either choose to go to Copenhagen Central Station (directions can be found above) or Nørreport.
Nørreport can be reached by taking bus 1A from Esplanaden (Store Kongensgade) towards Avedøre Station and get off at Østerport Station. Then take the S-train in the direction Køge, Høje Taastrup, Frederikssund, Farum or Hillerød (lines A, B, C, E or H) and get off at Nørreport Station.
You will need a valid 2-zone ticket. The price is 24 DKK.
You can also walk to Langelinie and take a Hop on Hop off bus, that among others stops at Nyhavn, Amalienborg Palace, Tivoli etc. The price is 195 DKK. Read more about the Hop on Hop off bus here.
You are already in the city center, but if you do not want to walk, a taxi to e.g. Nørreport will cost approx. 175 DKK on week days and 200 DKK in the weekend and on bank holidays. The duration of the trip is approx. 10 minutes.