We are planning a late August/Early September trip to Japan to catch the end of the Mt Fuji climbing season. Other than that, we have between 6-13 days to spend in the country. We plan to spend a few days each in Osaka/Kyoto/Nara and Tokyo. However, I don't really know what there is to do BETWEEN these cities. If you have any suggestions for out of the way places between these two megacities, that would be much appreciated.

Hakone is lovely.

I second Hakone

As a side note - prepare for very, very hot temperatures. I've visited Japan in August and it was so hot I got a heat rash for the whole trip. During our last trip, we went in late September and the heat was also pretty extreme for some of the days. Pack nice light clothing, hats, portable fans, etc to make your trip more comfortable.
I also agree that Hakone is a fun side trip. Also make sure to take a trip to Nara to visit the deer!

Thanks- it's going to be somewhat extreme (0 C at Mt Fuji's summit crater, ~30ish in the cities) so we'll prep accordingly.

Nagoya Shinkansen museum.

I'll say this everyone else who's travelling to Japan, join the Facebook group Japan travel planning. Total lifechanger for when we went last September

Consider Hiroshima.. beatiful city - with lots of sad history which they turned into something worth seeing in my opinion.

Just a note, the Tokyo Olympics and Paralympics are happening end of July and throughout August. It might be busier in certain areas than usual as people would like to do some sightseeing after the games as well.

Thanks for the reminder- I will be in Japan near the end of the paralympics and will not be in Tokyo area at that time, so I'm hoping it will be relatively quiet.

Check out Japan Foods'n Booze based out of Oiso. Manami offers great experiences and such a fun time!

Ive been to Japan twice (2012 & 2017) In 2017, I went in April and we visited Tokyo, Kanazawa, Takayama, Kyoto & Osaka. We go ourselves the JR Pass and went everywhere we could in 2 weeks. Originally on my itinerary, we wouldve stopped in Nagoya for a baseball game and done a day trip to the Yokohama, Endoshima, Fujisawa area. Another alternative is to go west to Kobe & Tottori (they have sand dunes). Nara, Kiso Valley and Naoshima are also on my list as well of course hiroshima. You should check to see what types of festivals are happening around the time you will be there. These are all options to check out to avoid the bustle that Tokyo will have coming off the olympics. You can check out my itineraries on my travel blog http://boketto.rosannau.com/