Shinjuku and Kawaguchiko are connected by expressway bus.
https://www.highwaybus.com/gp/inbound/index?lang=EN: Select "Tokyo" and "Yamanashi" then search.
Shinjuku Expressway Bus Terminal is in Busta Shinjuku,
which is near the New South gate of (JR) Shinjuku Station.
JR Narita Express trains from NRT
arrive at Track 6 of Shinjuku Station.
You go up to the bridge,
and pass through the New South exit gate.
Then, you walk to Busta Shinjuku and go up to the 4th floor.
- Maps of (JR) Kyoto Station
https://www.westjr.co.jp/global/en/timetable/pdf/station_kyoto.pdf: On the first / second map, the north is to to the bottom / the lower right.
Arriving at Kyoto Station by the Shinkansen,
you go down to the concourse
and pass through the Shinkansen Central Transfer Gate,
where your Shinkansen reserved-seat ticket is collected
and your basic fare ticket (to Kyoto City Area) is returned.
(You keep this basic fare ticket until Saga-Arashiyama.)
Then you cross the bridge
(above the conventional (= non-Shinkansen) tracks),
getting down
to around the joint of the two platforms for Track 0 and for Tracks 30 to 34.
Haruka trains for KIX mostly depart at Track 30.
Trains for Saga-Arashiyama depart at Track 31, 32 or 33.
You can visit the JR Ticket Office beside the Central Gate,
being still inside the gate.
In any of the two cases (a) and (b) shown below,
to make a stopover at Kyoto Station on your way to KIX,
you need tickets separately until and beyond Kyoto.
(a) You do not reserve Haruka seats.
The Haruka discount ticket (Kyoto Area, 1600 yen) is suitable,
which can work as a one-day, one-way ticket
to get from Saga-Arashiyama or Kyoto to Kansai-airport.
https://www.westjr.co.jp/global/en/ticket/icoca-haruka/(b) You reserve Haruka seats.
A basic fare ticket and a reserved-seat ticket are necessary.