Kyoto's problem is not finding things to do but narrowing them down. Seventeen sites in the city are inscribed together as a single UNESCO World Heritage entry, and you could spend a week on those alone without seeing everything. A sensible first trip would include Kinkaku-ji (the Golden Pavilion), Ginkaku-ji, Ryōan-ji with its famous dry stone garden, Kiyomizu-dera on its hillside platform, and the vermilion-gate tunnels of Fushimi Inari.
The east side of the city — Higashiyama — is best explored on foot: the lantern-lit lanes of Ninen-zaka and Sannen-zaka leading up to Kiyomizu, the Yasaka Pagoda on the skyline, the great Zen monastery of Nanzen-ji at the foot of the hills, and the bamboo grove of Arashiyama on the western outskirts.
Nijō Castle, with its famous nightingale floors, is the main surviving Tokugawa-era residence. The Kyoto Imperial Palace and the two great imperial garden villas at Katsura and Shūgakuin reward the extra planning required to book them.
For a quieter day, the Philosopher's Path walks you between Nanzen-ji and Ginkaku-ji along a canal lined with cherry trees, with a string of small temples (Hōnen-in, Eikan-dō) just off to the sides. Out toward the western foothills, the moss garden of Saihō-ji and the more accessible moss garden of Sanzen-in in Ōhara are some of the country's finest. For unforgettable Kannon sculpture, head to Sanjūsangen-dō and its 1,001 statues.
After dark, the best thing to do is simply to wander the back streets of Gion — the lantern-lit Hanamikōji and the willow-edged canal of Shirakawa — and watch the lights come on in the wooden teahouses. The narrow alley of Pontochō on the far bank of the Kamo is the parallel choice for an evening's wander.