About 90 minutes west of Osaka on the Tokaido/Sanyo Shinkansen (Bullet train) is Hiroshima. The main reason to go there is to visit the Peace Park and Museum, which are only a few stops from the JR station on the tram. You could do this in about two hours.
Hiroshima a bomb dome image Panoramio by Tomi Mäkitalo
Back at the station, you can catch a local JR service to Miyajimaguchi station (about 20 minutes), from where it is a 5 minute walk to the ferry which crosses to Miyajima island. If you have a JR pass, it is valid on one of the ferries, the trip taking about 10minutes. As you approach the island , you will see the famous red floating torii gates of Itsukushima shrine. This place is a very popular tourist draw card for foreigners and Japanese, and you can easily spend a couple hours wandering around, avoiding the attentions of the local deer. There are cable cars, restaurants, souvenir shops, temples and the shrine to explore if you have more time.
Should you wish to stay overnight, accommodation in ryokan (traditional inns) is available, though not cheap. You may prefer to stay a night in Hiroshima instead.
Torii- image Panoramio Masken
I have got up very early in Kyoto and visited both Hiroshima and Miyajima in a (long) daytrip, but if you can afford the time, spend a night or two out there, and explore the city a bit more. The castle ruins are interesting and within walking distance of the Peace Park, and you will have time to break your journey to or from Hiroshima at Himeji.
About 30 minutes from Shin-Osaka station or an hour from Hiroshima, is Himeji, home of one of Japan’s top 3 original castles.
JR Himeji station is only a 10 minute walk from the castle, which you can see from the train. As with most Japanese stations, there are lockers for your luggage, so you could stop here on the way to your next hotel, or just make a half day trip. Allow a couple of hours to explore the castle, which is fascinating. The park is beautiful in cherry blossom season.