Rural Japan is absolutely beautiful, I've travelled extensively there. The bullet trains are wonderful, but try going somewhere only the local trains go!
If you like biking, the Shimanami Kaido is unsurpassed.
https://www.japan-guide.com/e/e3478.html
Goes over multiple huge suspension bridges, grade-separated from the car traffic. In between the bridges you can explore the small towns on the islands in between.
To get there, go to Onomichi
https://www.japan-guide.com/e/e3475.html
which is a small city well worth visiting in and of itself. I recommend staying the night there, if you stay overnight in places that tourists usually day-trip to you get to see another side of the city. And when you're ready to hit the bike trail, you can rent bikes right there.
Further east, the Yamanobe Road is beautiful --
https://www.japan-guide.com/e/e4189.html
imagine walking all day through the set of My Neighbor Totoro, brought to life. Small towns, temples and shrines, mountain paths, small cities, and you'll walk by plenty of farms too. Easy to get to from Kyoto or Nara.
Further off the beaten path I recommend the Shikoku Henro Trail:
https://shikoku-tourism.com/en/shikoku-henro/shikoku-henro
If you have two months, walking the entire thing is the experience of a lifetime. But you can also spend just a few days (or one day) on the trail. I recommend starting at Temple #1 (they're numbered, this will be in Naruto, in the NE corner of Shikoku) and hiking for a day or a few days.
Japanese onsen -- hot springs -- are a great experience, look for a rotemburo, or open-air bath. Unfortunately most of these tend to be very off the beaten path, pretty well inaccessible by mass transit. But if you find one in your path, go!
You and your son are in for a treat. Enjoy!