[sarasi]

3 and a half weeks is more than enough to see Tokyo, Kyoto, Osaka, Hiroshima and Miyajima- that's actually a standard 10 day itinerary, so 3.5 weeks won't be rushed at all.

Just so you won't be disappointed, be aware that Mt. Fuji is not always visible as it is very often covered in cloud in any season, or invisible from further away due to heat haze in summer. You could be right next to the mountain in Kawaguchi-ko and not even know it's there. Your best chance of seeing it is early morning before the clouds form on the mountain.

No need to take tours to Mt. Hakone or Kawaguchi-ko, just take a train/bus out there and either do a day trip or stay overnight. Follow the instructions for how to get there in your guide book- and if you don't have a guide book, like Lonely Planet or Rough Guide, get one now and start reading it because you will really need it while travelling.