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.
From Malmö in Sweden you can go by train or car using the newly built bridge. You can also take a ferry if you like.
Malmö - Copenhagen (turn and return) by train costs around 140 SEK (£9).
In my opinion Copenhagen have the best airport in the world. And you can take the same train in to the city center, or cross the channel over to Sweden.
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.