The easiest way to get around in Copenhagen is to use the S-tog (Interurban train). It is cheep and reliable. You buy a 24-hour ticket at the Central station or at HUR´s sales boot (A big black building at Rådhuspladsen). The ticket also get you around North Zeeland From Roskilde in the south to Helsingør in the north. You can use bus, train or the new Minimetro to Amager. A island just west of Copenhagen.
Train from Hamburg Hbf to Copenhagen central station. €127 return including Auslandszuschlag (German rail fee for cross-border journey), seat reservation, Twentyticket rate.
You'll need to purchase a Danish foreign supplement fee on the train. (grrr!)
I booked the tickets from the German railsite and I got them about a week-10 days later.
Subway. And amazingly there's nobody checking the tickets. So you have to punch in accordingly. But once you got caught by the inspector not punching in the right fare, you are in a big trouble for sure.
This is the old local train system of Copenhagen - there's a lot of stations all over the downtown area.
For destinations and ticket system have a look at my other tip with S-train and Metro map.
Denmark has a wide network of trains. The station is in the heart of the city.
Since I live in Sweden the easiest way to travel to Copenhagen is with train and this is what the train station looks like, very big and nice.