Miyajima Island (Itsukushima) is the obvious first choice — the Itsukushima Shrine and its floating torii gate are one of the classic images of Japan, and the island itself is compact and walkable. The mountain of Misen above the shrine can be climbed by ropeway or on foot (about 90 minutes up). Allow a full half-day minimum; a full day if you want to climb the mountain and have lunch on the island. Getting there: Hiroden Line 2 to Hiroden-miyajimaguchi, then the ferry (total about 80 minutes from the city centre).
Iwakuni (45 minutes by JR San'yō Line) is best known for the Kintai-kyō, a five-span arched wooden bridge built in 1673 and rebuilt to the original design in 1953. The swooping curves are extraordinary — the bridge was built without nails, using only wooden joinery — and the surrounding park on the river bank is lovely for a picnic. Across the bridge is a cable car up to Iwakuni Castle and wide views over the Nishiki River. Day-trip territory, best combined with a walk through the former samurai quarter (bukeyashiki).
Ōno-Hara (Hiroshima's "oyster coast", roughly 20km west along the bay) is where most of Hiroshima Prefecture's oysters are farmed. Several small seafood restaurants near the boat docks offer grilled and raw oysters at prices considerably lower than in the city — the season runs October to April.