Shimanami Kaido Bikeway
This 70km bikeway connects Onomichi with the islands of Mukaishima, Innoshima (for the pirate Suigun Castle), Ikuchijima (for Kosan-ji), Omishima, Hakatajima, and Oshima, ending in Imabari on Shikoku. You're welcome to walk or jog any length of the bridge, but cycling is the best way to go. The route consists of wide dedicated lanes for bikes and pedestrians, typically alongside but separated from the car roadway. Special bike lanes have been created to provide reasonable approaches to each of the modern bridges connecting the islands.
Bikes can be rented or dropped off at any of the 14 bike depot locations along the route (Y1000 to rent, Y1000 deposit, deposit forfeited if you return to a depot other than the one you rented at). Bike depots have a variety of bicycles including 8-speed road bikes, 3-speed "upright" bikes, and electric assist bikes (electric assist must be returned to the original depot it was rented from). Helmets are available but sizes may be limited. The bike depot at the Onomichi port (7AM-6PM, +81 0848-22-5332) is located in the parking structure to the right of the Green Hill Hotel, located opposite the JR Train Station. On weekends, you should plan to rent your bike before 10am if you are starting from Onomichi.
There is excellent information including a printable map at this web site: http://www.go-shimanami.jp/global/english/guide/ You can also get a larger map (recommended) at Onomichi port or any of the bicycle rental places. This route is road cyclists heaven. The map is nice to have but the route is marked on the road with distance markers every km. Turns are shown with arrows and the entire route has a blue stripe on the road. Follow the stripe and you are on route.
Take the JR Sanyo train to Onomichi Station. The station is 20 minutes from the Fukuyama Shinkansen stop (the accessible frequent stop between Hiroshima, 25min, and Okayama, 15min). The Mihara shinkansen station is about 5 minutes closer, but it is served almost exclusively by the slowest, Kodama, trains, so if you are starting in Hiroshima, Mihara might be a good option, otherwise aim for Fukuyama, probably via the Sakura service.
You should note that there is also a Shin-Onomichi shinkansen stop, but it is rather far from town and difficult to get to Onomichi station (at the center of the city), so it will almost certainly take longer than the Fukuyama option.
If you don't have a JR Pass and are trying to save money (or are staying locally) It may also be reasonable to take the local train all the way from Hiroshima (85-100min/¥1,490) or Okayama (80min/¥1,320).
From Matsuyama (the largest town on this side of Shikoku and home to a castle and a few other attractions), the limited express Shiokaze/Ishizuchi takes about 35 minutes to Imabari. The Shiokaze continues to Okayama (125min) and the Ishizuchi splits off to go to Takamatsu (110min).
There is also a Hashihama station slightly closer (6km) to the first bridge and 5 minutes from Imabari station. Unlike Imabari station, there is no bicycle service/rental center at this station, but you can walk to the Sunrise Itoyama center (near the base of the bridge) in about 20 minutes.
The Green Hill Hotel is also the location of the ferry connecting Onomichi with Mukaishima, Innoshima, and Ikuchijima/Setoda. Serious bikers intent on completing the route in a day could do so, but the route is best enjoyed over the course of two days or in one day by taking a ferry from Onomichi or Mihara to Setoda (bypassing the two most rural islands on the route). Ferries leave Onomichi once an hour starting at 7:20 in the morning; the trip to Setoda is 40 minutes. Tickets are available in the ferry terminal behind the Green Hill Hotel.
The main route is generally flat except for reasonable and well-graded approaches to the bridges and one large hill on Oshima Island. A small toll is required at the entrance of each bridge, ranging from Y50 per rider to Y200; a discount pass is available at the bike depots if you will be riding across all the bridges.
The "recommended" main route distances are
Imabari station is a further 6km
There are a limited number of buses between Sunrise Itoyama and the Imabari JR Train Station; buses depart Sunrise Itoyama at 8:55am, 10:10am, 1:40pm, 4:10pm and 6:15pm. If traveling in the direction of Onomichi to Imabari, Kosan-ji Temple and Hirayama Ikuo Museums on Ikuchijima Island are about 4 km along the main route. On Omishima Island, in order to visit the Omishima Museum and the Oyamazumi Jinja Shrine Treasure Hall you must make a round trip (15 km total) off the recommended main route. If taking two days for the ride, Tatara on Omishima is a good place to spend the night before/after this side trip to Oyamazumi Shirine.
Onomichi is famous for its many temple.
Hiroshima city is 45 minutes from [Onomichi] by train, and it is perhaps most famous for the peace memorial and museum.
Miyajima, perhaps the most iconic site of Japan, is a further 30 minutes from [Hiroshima]
Imabari is a castle town.
Matsuyama, one of the main draws for visitors to Shikoku, is 35 minutes from Imabari.
Shikoku is known for it 88 temple pilgrimage, numerous well-preserved castles, and many areas of natural beauty.