Kyoto National Museum: This is a great museum for exploring the art and treasures of the area, temples and shrines. The cultural arts that flourished in Kyoto spread to the rest of Japan, so that much of what we call Japanese art and culture have origins here. It has Buddhist and ink paintings, textiles and Heian-period sculptures.
Kyoto Museum of Traditional Crafts and Kyoto Handicraft Center: This museum has great displays of Japanese handicrafts plus a large store onsite with many items for sale.
You might also check out The National Museum of Modern Art in Kyoto. The collection contains Japanese-style painting, Western-style painting, prints, photography, sculpture, ceramics, textiles, metalwork, wood and bamboo works, lacquerware and jewelry.