Mt. Misen
If you have energy to spare, Mt. Misen offers gorgeous views of the island and sea. The hike takes about an hour, depending on rest stops along the way. Look for the signs for the ropeway, and when you reach it, just keep going. There is a less used, slightly more strenuous and more rewarding route called the Omoto Pass that begins in the mossy park near the aquarium. Those with less energy or inclination can cheat and take the ropeway to an observatory near the top and hike from there ¥1000 one way, ¥1800 round trip. The ropeway closes at 5PM last car down 5:30. It is a 30 minute walk from the end of the ropeway to Mt Misen, you have to climb about 150 height meters after going down the first ten minutes.
Momijidani Park
The ropeway station is in the middle of Momijidani Park, but the park is worth a stroll even if you're not headed up. Known for its autumn colors, there are quite a few quiet little temples to explore along the way, and many of the island's hotels are tucked away here.