August 18 arrival in Kansai airport
August 19 Kyoto
August 20 Kyoto to Osaka and back
August 21 Kyoto to Mt Fuji Climbing for 5 people)
August 22 Mt Fuji to Kyoto
August 23 Stay in Kyoto
August 24 Kyoto to Nara and back
August 25 Kyoto to Suzuka Circuit and back
August 26 Kyoto to Kobe and back
August 27 Departure
Comments
Note however that you cannot get all the way to Mt Fuji by JR - you can take the bullet train to Mishima or Odawara and then catch a bus the rest of the way.
If climbing Mt Fuji, this info can help:
https://www.japan-guide.com/e/e6901.html
For Aug 26, I suggest you spend half a day in Himeji and half in Kobe. Himeji has the nation's finest old surviving castle, and the Kokoen Garden is very beautiful.
https://goo.gl/65ZtRi
You can check on train info on Hyperdia - you can see regular fares, routes, and schedules.
http://www.hyperdia.com
Under "More Options", with a full JR pass, be sure to uncheck the Nozomi box since you can't ride that train using the pass.
Also, you can look at some previews of some of the best places to see in Japan on:
https://www.youtube.com/user/thejapanfaq/videos
There are also loads of other tips on
http://www.thejapanfaq.com/the-best-japan-travel-tips/
Best regards.