Frequent daily Shinkansen run between Tokyo and Kyoto (2 hours, 40 minutes). The one-way fare, including charges for a reserved seat, is ¥13,220. Train service between Osaka and Kyoto (30 minutes) costs ¥540 one-way. From Shin-Osaka Station you can take the Shinkansen and be in Kyoto in 15 minutes; tickets cost ¥1,380. You may use a Japan Rail Pass on the Hikari and Kodama Shinkansen.
The Keihan and the Hankyu limited express trains (40 minutes each) are less expensive than the JR, unless you have a JR Pass. The one-way Osaka-Kyoto fare is ¥400 or ¥460 on the Keihan Line and ¥390 on the Hankyu Line. They depart every 15 minutes from Osaka's Yodoyabashi and Umeda stations respectively.
In Kyoto the Keihan Line from Osaka is partly underground (from Shichijo Station to Demachi-Yanagi Station) and extends all the way up the east bank of the Kamo-gawa to Imadegawa-dori. At Imadegawa-dori a passage connects the Keihan Line with the Eizan Railway's Demachi-Yanagi Station. The Eizan has two lines, the Kurama Line, running north to Kurama, and the Eizan Line, running northeast to Yase. The Hankyu Line, which runs to the Katsura Rikyu, connects with the subway at Karasuma Station. From Shijo-Omiya Station the Keifuku Arashiyama Line runs to western Kyoto. JR also runs to western Kyoto on the San-in Main Line.
