Hello Yvonne,
Both Kawaguchiko and Hakone are amazing to visit. I think Kawaguchiko is somewhat more popular when it comes to viewing Fuji. Though Hakone offers more in terms of sightseeing and activities.
The advantage Hakone has, is that it has a Shinkansen station that is directly on the route between Kyoto and Tokyo. This makes for easy access. The station name is Odawara.
Going from Kawaguchiko - Kyoto basically means traveling from Kawaguchiko back to Tokyo and from there to Tokyo. The whole route takes about 5 hours and has multiple transfers. See here for an example. The JR Pass covers the route, expect for the portion between Kawaguchiko - Otsuki on the Fujikyu line. This route costs 1,140yen.
2.) Within Kyoto the most regular way of transport is by bus. These go pretty much everywhere. You can also purchase a handy bus ticket for 500yen per day that allows for unlimited travel. You can also use the JR Pass for a limited extent in Kyoto to visit Fushimi-Inari, Arashiyama and Uji.
Hope this helps,
Daniel