A day trip out of Tokyo would indeed be nice to see more of Japan.
I just went to Japan for 10 days a week ago. We did Nagoya (Flying Blue promo out of Europe) to Tokyo (3 days), Nikko (incredible site) + Yokohama (biggest chinatown but disapointing at night as we did not go to the gardens) in one day, 3 1/2 day Kyoto (the true gem), one day Nara (no more is needed), Osaka is bit dull after Tokyo and Kyoto (people even seem rude not helping you out of the blue after Tokyo).
We also had a one day Tokyo-Hiroshima-Miyajima-Osaka train trip to see the Bomb museum and memorial sites + breath-taking Miyajima (best souvenir in Japan) which is 30 min. train + 10 min. ferry outside Hiroshima.
I would replace Osaka by a day trip on an early shinkansen to Hiroshima, 2 to 3 hours to see the site and museum + 4 hours to visit Miyajima (try to get on the mountain lift to see the interior see of Japan and even climb to the Mijajima highest point).
Japan Rail Pass is good but beware : Narita Express is not included I seem to remember and so are excluded Nozomi Shinkansen that run with very limited stops between Tokyo-Kyoto-Osaka. A Hikari shinkansen would just add half an hour I seem to remember. Avoid Kodama, those are REALLY stopping everywhere. Better to stand up in a non-reserved cas than seating in a reserved seat on a Kodama shinkanzen (that's the mistake we made the first day between Nagoya and Tokyo... to add to MAD-CDG-NGO...)
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