As far as day trips go, traveling to see Mt Fuji such as going to Hakone is easily done. But going to Osaka/Kyoto, not to mention Hiroshima and back within a day is not prudent. It would be far, far better to stay the night there.
In Kyoto you could spend weeks there and not see everything. The three best sights though are the Kinkakuji Temple, Kiyomizudera Temple, and Fushimi Inari Shrine. Fushimi is the most time consuming; you could spend a couple of hours to over half a day if you want to romp over all the trails.
The Fushimi Inari Shrine is something people of all ages would love very much. And the Otagi Nenbutsuji is one of the most humorous and whimsical places you could visit in Japan - well worth visiting.
Close by, Nara is often neglected by some tourists, which is a shame. Missing the Todaiji in Nara is like going to India and skipping the Taj Mahal. Nara Park has a lot of great places. A few other great places missed out even by those who go to Nara though are the Isuien and Yoshikien Gardens. They are gorgeous and not crowded at all.
In addition, there are other numerous great places you can visit in the area. Okayama is not far and it is famous for its castle, plus Korakuen, one of Japan's Top 3, and also the Handayama Garden. If you'd like to get off the beaten trail for a while, the half day hike up to Konpirasan in Kotohira is stunning. Going to Takamatsu is also nice - it has one of Japan's finest gardens also, called Ritsurin.
Hiroshima also has a number of good places. Miyajima is the most famous, and aside from Hiroshima itself, Onomichi with its temple tour and Kosanji Temple is spectacular. Just beyond Miyajima also lies Iwakuni with its historical bridge and castle on top of the mountain.