With your short time, it may make sense for you not to go to Tokyo. Though it has its attractions, Tokyo is not to Japan as London is to England or as Rome is to Italy -- the really central attraction culturally and historically is the Kyoto/Nara area. The main problem in skipping Tokyo might be that it makes including Nikko more difficult.

Instead of including Tokyo, you might consider going up in the Japanese Alps to the Takayama and Matsumoto area or maybe all the way over to Kanazawa, a very rewarding trip that can be easily done by trains and buses. It will also be cooler up in the mountains.

In Kyoto if using public transportation I found it very useful to stay as close as possible to the central train station -- most of the bus and train transportation converges there.

Another suggestion: find a list of traditional festivals (Matsuri) and see if there are any it would be convenient for you to take in.

In my opinion every human being alive ought to visit the Peace Museum in Hiroshima.