I would recommend opting for Kaiseki in Kyoto. I have been to a few places where the food is excellent, the atmosphere relatively informal and the prices low.

They are as follows:

Gion Tanimoto
Gion-machi 570-121; Tel: 075-551-8011
Good food and friendly staff. Lunch is Y5,000; dinner Y8,000

Giro Giro
Kawabata-dori San-jo agaru 2-177-1; Tel: 075-533-1155
Courses from as little as Y2,000 served at a single counter by punk chefs. Difficult to get reservation.

Kenninji Gion Maruyama
Komatsu-cho 566-15; Tel: 075-561-9990
Sister to a grander restaurant down the street, but the food is excellent. Lunch is Y8,000; dinner in the region of Y20,000.

Get your hotel concierge to make reservations as you shouldn't expect anybody to speak English at any of these retaurants (although the waitress at Kenninji Gion Maruyama was telling me she was learning French for the legions of Gallic customers that descend on the restaurant!)