Regarding cycling, do you intent to do a more sporty version of cycling, or a more leisuerly?
Definitely both is possible in that area.
For the more sporty version you could simply ride up to the 5th station on Mt. Fuji. Or once around Mt. Fuji (that's about 100 km with a serious ascent from 400 m to about 1100 m and some more ascents). Or a shorter version to "just" ride from Gotenba station up to Yamanakako. About 25 km, but with a serious ascent from about 200 m to 1200 m.
If you are not in for ascents, but still want a sporty variant, you could do the cycle round of the 5 lakes, that's about 100 km, but it is all between about 800 m and 1000 m elevation.
If you are not in for the sporty variant, you could cycle around only one of the lakes.
I have to admit, I never have cycled in that area, so my knowledge is from books. If you are interested in any of the above, let me know.
If you are, like me, more a leisurely cyclists, I can warmly recommend you two cycling experiences a bit outside of Tokyo:
1) Start in Shinjuku and follow Kanda river until Kichijoji. Then continue on to Koganei park (where the Edo Tatemono museum is, which is also very nice) and then find the entrance (see map below) to the cycling road that leads straight (I mean REALLY straight) to Lake Sayama. You can then cycle once around the Lake and ride back on the same way to Tokyo. (Or there are alternatives, e.g. to Tamagawa josui and follow there the channel back to Koganei park.). In total from Shinjuku that's probably around 55 km, no heavy ascents. You can cut it short by starting the ride later (e.g. at Kichijoji). The problem with riding bicycle in Japan is that you are not allowed with a bicycle on a train UNLESS it is in a bag. So if you are a road cyclists, you might have a bag already and that's fine. If you want to ride however on a rental bicycle (there is a good system now in Tokyo), you cannot bring it on a train. http://bicycletraveljapan.blogspot.com/2017/06/ride-to-sayama-lake.htm...
(sorry, no big text, just the map)
2) the other ride is down a river towards Enoshima. You can start for example in Machida, and just "jump" on the cycling road that runs along the river towards Enoshima. If you want to cycle less, then you start farer down the river. The nice thing about this cycling experience is that it is all on cycling lanes and you cannot really get lost, because you are constantly riding alongside a river and you end up in Enoshima. Again the transport of the bicycle to (and back to) the starting point can be a problem, as you are not allowed on trains, unless bicycle is in a bag.
I loved to end that ride in Kozashibuya and get a good soak in the local onsen (on the 5th floor of a supermarket, https://www.ousama2603.com/shop/kouzashibuya/
) with views on Mt. Fuji!
For more onsen ideas, here my list of favorits: http://bicycletraveljapan.blogspot.com/search/label/Onsen+Practicaliti...
Here a link to most of the cycling I did in Kanto area: http://bicycletraveljapan.blogspot.com/search/label/Kanto+Ride
All quite relaxed. No big ascents.
Enjoy your second trip to Japan!