This is a nice somewhat tourist-free island that is a good overnight stop. You can take the ferry from Kobe directly and there is a place to stay right by the ferry terminal on Shodo-shima. Bike rental is available right there. Another way to get to a different part of the island is to take the JR to Himeji. At the bus terminal across the street from the station take the bus to Himeji port. At Himeji port there is a ferry to Shodo-shima which takes about an hour and a half. There is a bus service on the island but on weekends it is not too frequent but you can get anywhere.
There is a very reasonable and beautiful inn in Ikeda. Monkeys and interesting rock formations in the mountains in the centre of the island. Just about anywhere you go there are good views and just very pleasant hiking, biking or even bus riding. Read the sentimental Japanese novel “Twenty-Four Eyes” before you go.