The restaurants in Kyoto are excellent.
I enjoyed trying local places in Gion and in the Downtown area. Kane-yo on Shinkyogoku-dori was a good place to try unagi or eel. It is very casual and has been there for over 100 years. I also enjoyed the Kyoto-style restaurant Tagoto, across from Takashimaya.
The Tawaraya: The famous Tawaraya ryokan offers a glimpse of the time-honored Japanese way of life. Still centrally located in Kyoto on a quiet street, it has been operating for over three centuries and is classed as one of the finest places to stay in the world. The food, excellent. I ate one dinner in my room and the other at their separate, very attractive restaurant called Ten-you a short distance away. They serve a full course dinner, Tempura Kaiseki, which included tempura and small dishes of Japanese delicacies. They also have a lovely tea room around the corner which was a great place for a late afternoon snack.
You will find small tea rooms, local sushi bars and other great choices. At the Hotel Okura there were also good restaurants in the attached retail mall below. I found a very good Udon restaurant for lunch.
Izusen: Located at the Daitoku-ji Temple. After a Tea Ceremony, make sure to have a lunch of shojin ryori, vegetarian Zen style cuisine, within the temple grounds. It was a wonderful experience.
Other restaurants getting wonderful reviews include Kitcho, Muromachi Wakuden, Kinmata and Kikunoi.