I have no kids but the itinerary my tour group followed (2 weeks) had some nice variety in it that kids would probably like too. Here’s the list

We arrived in Fukuoka and worked our way up to Tokyo where we took the plane back. If you’re limited in time I’d drop Fukuoka, Kurashiki, Koraku-en and Naoshima although kids would probably enjoy this art island too. We mostly travelled by bus but Japan has a very good train system.

• Miyajima (sacred island with dears and monkeys)
• Hiroshima (if the museum is too much for kids, the memorial park is really nice and there are tons of Japanese schoolkids on trips there too)
• Himeiji (samurai castle)
• Kyoto (“old Japan”, temples, especially Fushimi Inari)
  
  • trip to Nara (nice park with shrines, dears, the buddha’s nostril thing)
• and then a Shinkansen trip
• Tokyo (“new Japan”, Tsukiji fish market is cool if they’re not squeemish)
  
  I haven’t been there yet but Nikko looks really fun to me too (hot springs with monkeys)