We got the train from Malmo over the fairly new bridge.
Everything that trains in the UK often aren't, clean, punctual, modern and convenient - it took us into the heart of Copenhagen and let us spend a great day out there.
If you go from Sweden, go to Hesingborg and take the ferry to Helsingör. It only takes 15 minutes. Now, you should also try the bridge between Malmö and Copenhagen. It is also nice to tale the ferry to Denmark from Norway.
You should bike around Denmark! It is also ok to go by car or train.
Ferries ply the waters of the narrow channel separating Helsingør (Denmark) from Helsingborg (Sweden) in less than 25 minutes. They're operated around the clock by Scandlines.