The easiest way to get from the centre of Copenhagen to Dragør is to take bus no. 350S (departs from Nørrebro Station, Nørreport Station and Christianshavn).
A one way ticket costs 36 DKK (January 2013) and the ride takes about 40 minutes.
I arrived at Dragør by car and we walked around the older part of the village, but didn't have time to rent a boat and explore the waterways.
The #40 bus from Copenhagen will get you there in about 40 minutes. There was a ferry from Dragør across the way to Sweden but there is now a convenient bridge if you want to drive over..
It couldn't be easier to get to Dragoer. Just catch a bus from Kastrup airport, Tårnby or Örestad train stations. There is also a bus all the way from the centre (see below).