This region in the west of the city has three well-known gardens all directly adjacent to one another. Take the #28 bus and get off at Tenryu-ji, which is the name of the temple containing the first. Buy a Y500 ticket on your left next to the temple, and make a circuit of the grounds, ending up at the Arashiyama Bamboo Grove.
The Bamboo Grove is a massive stand of the fast-growing grass, and the height of the canes filters the sunlight to a slightly eerie green shade:
At the top of the grove you will find the Okoshi-Sanso Villa, admission price Y1000, which, at a conversion of USD 14 is rather steep for a garden. However, most everything in Japan is expensive, and you find yourself eventually just giving a little shrug and paying up. The price does include a bowl of green tea and a Japanese sweet in the nice tearoom at the end. The garden belonged to a famous silent-film era star, Okochi Denjiro (1898-1962), who built the grounds over a 30 year period on the south side of Mout Ogura. There is a nice overlook of the city of Kyoto.
Gyoen National Garden
Located in central Kyoto, this is essentially the grounds of the Imperial Palace, Japan's seat of government from 794 to the early 1600s. It's open during daylight hours, and you can tour the palace itself, but only with a guided tour (available in English).
Though small, this delightful garden was the best I saw in Kyoto. Affiliated with the Higashi Hongwan-ji temple, it is an even shorter walk from my hostel than it is from the temple further west.
Remember, you can click on any photo to see a full-size version.