There are daily planes from Cairns to Horn Island. From Horn Island you can get a taxi to the wharf, and from the wharf you can get the ferry over to Thursday Island commonly known as "T.I." the ferry cost $10 each way. There are regular boats from Thursday Island to most other populated Islands, but you will need to get permission from the appropriate indigenous land council beforehand.
You can also get in from Papua New Guinea if you have your own boat. If you do so, you will need to check in with the department of immigration once you land.
Peddell's Ferries provides twice daily passenger transport by boat to Thursday Island from Seisia, Cape York. Check their website for up-to-date timetables and fares. Tickets can be booked online.
Those intent on a road trip can get a barge from Seisia on Cape York.
