Best way to arrange transportation

We are a party of 8 going to JP on Arp 13-20. The plan is as follows:
4/14: arrive to Haneda airport- need transportation to go to a guest house in Shibuya.
4/15: to Haneda Osen and Mt Fuji- Does 2 day Haneda pass work?
4/16: possible going back to Mt Fuji or Nikko- What transportation needed?
4/17: 5 Lakes or explore Tokyo
4/18: go to Kyoto- Should I purchase JP rail to use Shikansen train?
4/19: stay in Kyoto-Osaka. If have time to go to Nara.
4/10: afternoon go back to LAX from KIX.
I wonder what is the best transportation should I get? Can I buy 1 way JP rail for Tokyo-Kyoto?
Can I get two two-day subway passes while in Tokyo and 2 day subway pass while in Kyoto/Osaka?
How about the trip to Haneda/Mt Fuji, what transportation should I get so I could get around town and to the bottom of Mt Fuji?
Thanks so much for your help. This is our first trip, so I have no idea.

Excuse me, but you are underestimating the travel time between places that you will barely have any time to really see anything.

Haneda to Shibuya: use Keikyu train to Shinagawa then change to JR Yamanote line to Shibuya.
Do you mean Hakone? Take Odakyu train from Shinjuku.
Nikko: either Tobu train fromAsakusa or JR.
Kyoto: JR Shinkansen.