From Kyoto to Kanazawa, take a JR limited express "Thunderbird".
If you come to Kanazawa directly from Kansai International Airport, take a JR airport express "Haruka" and transfer to "Thunderbird" at Shin-Osaka or Kyoto.
1. Timetables of "Thunderbird" between Kyoto (or Shin-Osaka) and Kanazawa
Monday-Friday
Saturday, Sunday and Holidays
Kyoto (or Shin-Osaka) to Kanazawa
Kanazawa to Kyoto (or Shin-Osaka)
2. Timetables of "Haruka" between Kyoto (or Shin-Osaka) and Kansai Airport
Monday-Friday
Saturday, Sunday and Holidays
Kansai Airport to Kyoto (or Shin-Osaka)
Kyoto (or Shin-Osaka) to Kansai Airport
3. Other travel routes in Japan can be searched on "Hyperdia"