10 Day Japan Itinerary

Hi, will be in Japan for 10 day trip end May - early June.
Appreciate advice on flow and mode of transport. Am getting confused by the many types of rails, buses etc.
Day 1 - Tokyo. Arrive in morning. Stay the night
Day 2 - Tokyo to Takayama (4days, 3nights)
Day 4 - Takayama to Hakone (3days, 2nights)
Day 8 - Hakone to Tokyo
Day10 - Homebound
We could especially advice on the best way to travel from Takayama to Hakone on Day4.
And, any suggestions on what not to be missed in Takayama and Hakone, please?
Many thanks.
Misc

How to travel between Tokyo and Takayama:
http://www.japan-guide.com/e/e5908.html

As for Takayama to Hakone, I recommend to take the Hida limited express from Takayama to Nagoya and then the Tokaido Shinkansen to Odawara (or Mishima) Station and catch a bus or train from there into Hakone.

And, any suggestions on what not to be missed in Takayama and Hakone, please?

I recommend not to miss the sights with two or more red dots. The rest I recommend to chose according to personal interest:

What to see in Takayama:
http://www.japan-guide.com/e/e5900.html

What to see in Hakone:
http://www.japan-guide.com/e/e5200.html