Hi there,
Assuming you use the JR Pass, the best route to take is to travel to the station Otsuki close to Tokyo which takes 85 Minutes. There are many trains here going each our so there is no need to worry about a timetable. From Otsuki you can take the Fujikyu line all the way to Fuji which is 1,100 Yen / One way and takes about 50 minutes, trains leave about every 20-30 minutes depending on the season.
The best way to travel on to Kyoto is to take the same route back to Tokyo and take the Bullet train from there. From Tokyo it is about 160 minutes to Kyoto. Trains leave every 30 minutes.
I would recommend using Hyperdia for your personal planning, it is a great resource that allows for very detailed planning. I would recommand reading our guide about planning with Hyperdia when you are yet unfamiliar with it.
There is a direct train between Kyoto and Osaka called the Shinkaisoku, this train leaves every 10 minutes and takes less than 30 minutes to reach Osaka from Kyoto or visa-versa. It is also covered by the JR Pass.
To Kansai Airport, you can take Airport Express from Osaka station or the Haruka Express from Kyoto/Shin-Osaka both trains are covered by the JR Pass.
I hope this helps!
-Daniel