20 Days in Japan

Hi,

A 14 day pass would serve you very well, activated from Narita Airport upon your arrival.

There are numerous half or full day trips you could take during your trip. Just beyond Hiroshima/Miyajima is Iwakuni with its iconic bridge and mountaintop castle. Then there is is the Akiyoshido Limestone Cave which like Iwakuni is also in Yamaguchi Prefecture. It's a 300 million year old wonder with giant underground caverns and and river flowing past you.
Also in Hiroshima is Onomichi with its temple tour and wildly ornate Kosanji Temple; it's so out there that it's been called something like Disneyland.

Between the Kansai area and Hiroshima there are also numerous good places off the beaten trail. Okayama is famous for its Korakuen and Handayama Gardens which are both beautiful. In Kagawa on Shikoku there is a gorgeous day hike at Kotohira to a nice mountaintop view over the whole area at the Konpirasan Shrine. And in Takamatsu, you can see the Ritsurin Garden which is one of Japan's best.

Closer to Kansai, seeing Himeji castle is also superb, and if you can stay a night at Koyasan for a very spiritual experience, it will be something you never forget. Kyoto and Nara are the jewels of the area though. You could certainly get away from the crowds a bit there also - take a hike up to Kurama, or explore Mt. Hiei.

Near Tokyo, take some day trips to Nikko and Kamakura/Enoshima. You could also get a regional JR Kanto Area Pass to explore the Kusatsu hot springs, Kamikochi and Izu Peninsula.

You should have plenty to see and do. To get back to Narita not using the pass, look into a 1000 yen bus.

Best of Luck.