Closest airport is in Split. From the airport you need to get to the port by taxi or bus. From there you can get a taxi boat of ferry to get to the Island. To buy the tickets and to see the schedule visit Jadrolinija or Kapetan Luka. Buy your tickets before you arrive because they are often sold out.
The best way to get around the island is by a car. You can rent a car in Split and then transfer it with a ferry or you can rent one on the island. You can also rent scouters or quads. If you don’t enjoy driving on narrow roads there is also a taxi option.