When I arrived in Japan for my first visit, I initially spent a couple of nights in Tokyo. The modernity and high style of the capital city was breathtaking—a total immersion in the near future rather than the present. A few days later, when I took the high-speed train to Kyoto, the city's classical charms and historic atmosphere—in bold contrast to Tokyo—were even more affecting.
Ancient Kyoto was once the imperial capital of Japan, and the city reigned as the cultural and artistic center of the nation from 794 to 1868. In Kyoto, most buildings are no more than three stories high. This means the mountains and hills surrounding the city are always visible, adding to its suspended-in-time ambiance. It's a wonderful walking city, with a river running through its center and green hills to the east, making it almost impossible to get lost. You'll see lots of people on bicycles, and it's a real tribute to the Japanese that rows of bikes can sit unlocked outside the entrance of a subway station without fear they'd be stolen. 
My first night was spent in Yoshi-ima ryokan in the Gion District. Ryokans are basically country inn-style accommodations, complete with wooden bathtubs, futons and low tables. I was given a yukata, a traditional Japanese robe, to wear. The room was wonderfully simple, with tatami reed mats on the floor, paper screens on the walls and a low black table in the center.
Dinner was served in my room. It was a multicourse meal comprised of artfully arranged dishes of tempura, salmon, miso soup, sashimi, pickled vegetables, white rice, green tea and strawberries for dessert. After dinner I was invited to a complimentary tea ceremony in the ryokan's teahouse, which gave me a fascinating glimpse into Japanese customs. The tea ceremony begins by entering through a two-foot-high door, the purpose of which is to humble everyone and strip away social rank. You receive the tea cup with "the good side" facing you. You cradle the cup in the palm of your left hand and turn it so the good side faces the host. The owner of Yoshi-ima joined us in the dimly lit teahouse and gave a talk about the history of the property and some of the lore associated with the tea ceremony. (My favorite part was when he finished his prepared speech and asked, "So how about that Obama? I hear he's winning the primary in Wyoming.")
I would recommend a one-night stay in a ryokan for anyone who has a sense of adventure and a sense of humor—you'll need to make instant attitude adjustments when you come face-to-face with Japanese traditions.
Kyoto's historic Gion District is a fascinating neighborhood of narrow streets lined with antiques shops and small restaurants. You can actually see real geishas walking the streets in the evening, en route to private parties. It's a great district to explore on foot, with the added bonus of its proximity to a number of Kyoto's major temples and shrines.
If you have more than one day in Kyoto, you might want to consider dividing the sightseeing up geographically: the east-side attractions on your first day, and the west-side ones on your second day. On the east side are such sites as Kiyomizu-dera Temple, Ginkakuji Temple (Silver Pavilion) and the Kyoto National Museum. On the west side are Kinkakuji Temple (Golden Pavilion) and Nijo Castle. Breaking it up this way will minimize your expenses criss-crossing the city by cab (Kyoto also has an extensive subway and bus system). You'll also have time to take a breath and really enjoy the various temple gardens. Kyoto is not a city to rush through. You can reduce the static in your brain by taking the time to contemplate a Zen rock garden, such as the one at the Silver Pavilion, or the ponds and fruit trees of the lovely garden at the Heian Shrine.