Shinkansen is the obvious choice from eastern Japan. Hiroshima station is on the JR San'yō Shinkansen; the fastest Nozomi trains from Shin-Osaka take about 85 minutes (¥9,750), from Tokyo around 4 hours (¥18,380). The Hikari and Kodama services take longer but are covered by the Japan Rail Pass. From Kyoto, plan on about two hours.
By air, Hiroshima Airport is 50km east of the city; limousine buses connect to the city centre in about 45 minutes (¥1,470). Budget carriers including Peach and Jetstar operate domestic routes, but for most visitors the shinkansen is faster door-to-door.
If you are travelling on a JR Pass and want to visit Miyajima, note that the ferry from Miyajimaguchi to Miyajima Island is operated by JR and is free with a valid pass.