The nearest international airport is Kansai International Airport (KIX). Once you arrive to Kansai International Airport, you can use train or bus to Kyoto city. Most convenient way is train. You can ride Kansai-Airport Express "HARUKA" to Kyoto station operated by west Japan railway company (JR West). It takes 1 hour 20 minutes from Kansai International Airport to Kyoto station. Please check the website of JR West.
If your final destination is Osaka International Airport (also known as Osaka Itami Airport (ITM)) which is mostly used by domestic flight, most convenient way to Kyoto city is bus.
If your destination of flight to Japan is airports in Tokyo (Tokyo-Narita International Airport (NRT) and Tokyo International Airport (Haneda Airport (HND)), you can go to Kyoto by Tokaido-Shinkansen train "NOZOMI" operated by central Japan railway company (JR Central). It takes 2 hours 30 minites from Tokyo station to Kyoto station (you should consider extra tranportation times from the airports to Tokyo station (about 1 hour)).
Kyoto is the one of the largest tourist cities in Japan. In the city, taxi, two routes of subway, and a lot of routes of bus survise are available. Please see the following information