Hi,
NIkko is about 150km north of Tokyo, and with a good connection (part way by bullet train) you can get there in about 106 minutes. Nara is about 42km from Kyoto and takes 44 minutes by express train (no bullet trains). Himeji is about 130km from Kyoto and by bullet train about 54 minutes (including changing trains at Shin-Osaka Stn).
For Nara, most of the best is at Nara Park. The Todaiji Great Buddha is a must see place. Also you can feed the deer around it, and see Kofukuji Temple, Kasuga Shrine, and the Isuien and Yoshikien Gardens are wonderful.
I recommend you check rail schedules on Hyperdia.
There is a common syndrome among Kyoto tourists who pack in as many temples/shrines they can in a day, and by the end of the day everything is just a blur - called getting "templed out". To avoid this it's better to break up the schedule with other things, and Kyoto has some good places as well. Visiting Nishiki Market is one (where many people go to buy their daily foods - it's a great window into cuisine and food culture), there is also a beautiful and mostly unknown garden between Nanzenji Temple and Heian Shrine called Murin-an located where the Niomon Street curves E to SE. Nijo Castle is another. Going to see the Shugakuin Imperial Villa is another good pick (prior reservations required), plus there is the Shogunzuka Mound with a city overview. On Kyoto's west side is Arashiyama, and there is a nice temple garden at Tenryuji, the famous bamboo grove is next to it, there is a monkey park on top of the hill facing that, and so on. And the Otagi Nenbutsuji is one of the most humorous and whimsical places you could visit in Japan - well worth visiting, even for kids.
Read some info on what the city has, and then plan accordingly.
Shin-Osaka is the bullet train station for Osaka. Osaka Station is for express and other trains. You could go to either, but the distance is not far and you don't save much time by taking the bullet train. Besides, to go somewhere else in Osaka, you generally have to connect to another train or subway, so Osaka station is the more common place to go to. Unless you want to whoop it up til way late at night (trains stop around midnight) then you might visit Osaka for the evening and go back to Kyoto for the next day.
Good luck!