The fastest connection from Kansai International Airport to Kyoto is by JR train: take the Haruka express train bound for JR Kyoto station. Trains depart every 30 minutes, and the journey takes about 75 minutes. The fare is 3,370 yen with seat reservation.
Alternatively, but still on the JR line, you can take the Kanku Rapid (Kanku Kaisoku) train to Osaka and change there to the Special Rapid Service (Shinkaisoku) for Kyoto. This is a bit more complicated, more time-consuming (about 2 hours), but much cheaper (1,840 yen).
There are also limousine buses running from Kansai airport to Kyoto.
NB: (1) There are two train stations at Kansai Airport: the JR station and the Nankai
station. For going to Kyoto, JR is much more convenient.
(2) Train tickets in Japan always have to be kept till the end of the trip. They are needed for getting out of the train system, and then disappear automatically in the exit gate. Thus, if you need proof of your trip for later refund, please insist on obtaining a separate receipt when buying the ticket.
(3) Express trains in Japan (= trains for which seat reservation is possible), like the Shinkansen superexpress or the Haruka express (see above) etc., have cars with reserved seats and cars with non-reserved seats. This is simple if you buy a ticket with seat reservation (shiteiseki) because the ticket says the car number. If you have a ticket without seat reservation, make sure you get on one of the cars with non-reserved seats (jiyuseki).
For train schedules in Japan look at: Japan Transit Planner.
NB: (1) There are other bus companies operating in Kyoto, like Kyoto bus, Keihan bus, Hankyu bus etc.
Do not mix city bus No. 17 with bus No. 17 of other bus lines.
(2) In Japan, one usually enters the bus at the back and exits in front next to the driver. One pays the driver when getting off. If you don't have exact change, coins and 1000 yen bills can be changed at the machine next to the driver.
Here are some webpages with information on public transport in Kyoto.
There is no tipping in Japan, neither for taxis, nor in restaurants.
Japanese electric current is 100 V. Most North American appliances will work reasonably
well on Japanese current. Plugs are also identical to North-American plugs.