Kyoto is an amazing city. I loved being there and really enjoyed it more than Tokyo. It was smaller, easy to get around and was a much easier walking city. There are many UNESCO World Heritage sites and the shopping, antiques, food, temples and gardens were really outstanding. I really like the Japanese culture and aesthetic.
In addition to the other posts about the shopping, temples and museums, I highly recommend:
Exploring Gion: This is the famous entertainment and geisha quarter. I liked walking though the smaller lanes where you might spot a real geisha or maiko (apprentice geisha) walking out of her residence.
Gion Kobu Kaburenjo Theater/Geisha: Try to attend a performance of apprentice geisha at the theater in Gion. It is somewhat touristy but interesting. I preferred to see real geisha in the streets. I also saw one with her companion at a restaurant during lunchtime. The contrast of this young geisha to the table of well-dressed, modern young women across from me using their cell phones was huge.
Shichigosan – An Ancient Japanese Tradition: Children’s Celebration in Japan (known as Shichigosan Blessing) is among the most colorful rituals in the world. Every November parents take their 3 and 5 year-old boys and 3 and 7 year-old girls, dressed up in their best traditional Japanese kimono and fashionable costumes, to local shrines to receive divine blessings from Ujigami, the Shinto guardian god of good health. This event is celebrated all over the country as a rite of passage. I was so fortunate to be there in November where I saw many beautiful children and their families at the major shrines. I hope you can see it for yourself!
Daitoku-ji Temple and Tea Ceremony: The Daisen-in Zen Garden was really lovely. At the Zuiho-in, I was introduced to the famous Japanese tea ceremony by one of the monks in a private ceremony. It was great to experience. You can then have a lunch of shojin ryori, vegetarian Zen style cuisine, within the temple grounds at Izusen.