Hi,
The order is not important as long as you are not backtracking or missing some event that you want to see.
Since you are planning on using a 21 day pass, you should do your best to complete your long distance travel within that period. Please be aware also that for some places like Shirakawago Koyasan, Hakone and Kamikochi, there is no JR rail access, so you will need to pay for the buses or non-JR rail there. For Koyasan, get a Koyasan World Heritage Ticket.
You could base yourself in Matsumoto to see the Kiso Valley and Kamikochi, and either Takayama, Shirakawago or Kanazawa for those places (but an overnight at a Shirakawago is a real treat).
I suggest you do Hakone as a day trip from Tokyo, just to keep things simple. Take the bullet train from Tokyo and get off at Odawara Stn., then to save money, buy a Hakone Free Pass. After seeing the area, return but go to Mishima Stn., then catch the next bullet train to Kyoto. The Hakone Loop Course is a common way to see the area.
For seeing Beppu, the fastest way is by going from or through Kokura in Kitakyushu. If you have the chance, seeing the city view from atop Mt. Sarakurayama especially in the evening is one of Japan's very finest, yet nearly unknown to the outside world. Beppu is famous for its hot springs, but its Hells Tour, plus the Takasakiyama monkey park/aquarium are all terrific.
Nagasaki is the most interesting city on the island and worth a couple of days as well.
Kumamoto is famous for its castle, plus beautiful Suizenji Garden as well as for Mount Aso, but the castle and some places at Aso are closed due to earthquake damage and volcanic activity.
For Fukuoka , there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
Don't miss the Nanzoin Temple which is one of Japan's finest sights.
Lastly, since you have so much time in Japan, you also might try to fit it a couple of days in Okinawa, which has some very good and unique places as well, You can fly there dirt cheap on low cost carriers like Jet Star, Peach or Vanilla Air.
Good luck.
