Kenninji, founded in 1202, is the oldest Zen temple in Kyoto, which is the center of Zen culture. It is considered to be one of the 5 most important temples in Kyoto. Enjoy the tranquility of the two Zen gardens, and be sure to explore the main Dharma Hall of the temple. Admission is free.
In this frantic world, Zazen in Kyoto is sure to induce a sense of calm and peace in anyone who is willing to try meditation. The ultimate goal of Zen is to obtain enlightenment but beginners will appreciate the sanctity of a natural environment where silence and solitude are valued. Zen centers accommodate visitors who are unable to sit cross legged on the floor with a chair, but it is best to check ahead of time.
Also consider joining a local scholar on a 3 hour walking tour exploring Shinto and Buddhist landmarks, including this temple: http://www.viator.com/eap/site/1/6684/tours/5411KYO76541