From Seoul, take the train (subway) 2 hours east and get off at Gapyeong Station. Then take the ferry to Nami Island, now called the Naminara Republic but the locals just call it Namiseom or Namisum.
From Gapyeong Station take a quick taxi ride to the ferry to Nami Island. Don't worry about finding a taxi - there are lots of taxis waiting for you right when you get off the station because that is the main reason why people go there. The ferry ride to the island is about half an hour and scenic as well.