Hi,
From Kyoto to Gotemba you'd have to change trains twice. It would be easier to go to Odawara, buy the Hakone Free Pass there, and start on your way to Hakone. You can get more info on the place here.
You can go to Hyperdia and just plug in your starting point and end point, plus date/time that you like. But be sure under "More Options" to deselect "Nozomi..." since you can't ride that on the Pass.
Actually, since you are already seeing Tokyo, you can largely skip Osaka during the day. After the temples of Kyoto close down around 5PM, you can zip over to Osaka for the evening. At night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg, and Abeno Harukas Bldg are wonderful.
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. And the Otagi Nenbutsuji is one of the most humorous and whimsical places you could visit in Japan - well worth visiting, even for kids.
Nara is often neglected by some tourists, which is a shame. Missing the Todaiji Great Buddha in Nara is like going to India and skipping the Taj Mahal. Nara Park has a lot of great places, such as Kasuga Shrine, Kofukuji Temple, and feeding the many deer in the area.
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. Taikodani is also a beautiful shrine in Tsuwano, Shimane.
Hagi Castle was actually delightful, and the preserved samurai district and Mori related temples a rarity. One of the very best is the Akiyoshi Limestone Cave, a 300 million year old wonder with huge caverns and underground river running through it.
All in all, a lot more to explore and enjoy.