The usual rail approach is JR Obatake Station on the Sanyo Main Line, just beside the Oshima Bridge. From there, buses and taxis cross onto the island, but services are thin enough that a rental car is the practical choice for most travelers.
Drivers reach Suo-Oshima by the Oshima Bridge from the Yanai side of Yamaguchi Prefecture. There are also ferries from Ihota on the eastern tip of the island toward Matsuyama, useful if you are linking the Seto Inland Sea coast with Shikoku.