Its probably too late by now, but i am living on shodoshima now. So perhaps my advice is more for future reference for anyone coming to the Island or if your to return someday.
The best option is to hire a car, but they all get booked out pretty fast, so its a great idea to book early for them.
Another very accessible and available mode of transport is the bike, but its rather restricting to the area you hire it from and its possible that you can only hire bikes from the Tonosho area.
Rental scooters might be a bit adventurous for heading up the Skyline road to Kankakei and choshikei but are probably an economical and convinient option, though ive never looked into it as i have a car.
As previously mentioned the bus network is pretty fundamental and can only take you so far. Once again i think you need to get on tour buses to head up to kankakei and the like. The local buses are quite pricey after a while too, especially if your heading across the island to eigamura or kankakei from tonosho/fukuda.
therefore, in conclusion, rent a car! Its the most comfortable, most likely relatively cheap, I imagine most the foriegn visitors here will be on a pretty leisurely schedule with a bit of time to soak it in (sunniest place in Japan i am told by the locals) so its worthy of a weekend.
If you enjoy driving the roads are very good to drive as well, with almost no traffic out of the towns on the likes of the kankakei roads or skyline. (watch out for monkeys). Finally with a car your free to explore. There are some very nice secluded, almost unknown beaches around the island if you look for them and certainly some stunning lookouts. You can see the seto Ohashi bridge from a few lookouts atop the island on a clear day/night.