Ryoanji Temple is the most famous Zen Garden in Japan. It dates back to the 15th century and is said to hide infinite truths in the riddle of rocks and sand patterns displayed there. It is very crowded due to it's popularity, but people are respectful and quiet. Most people just sit and look but you'll see a few foreigners trying to meditate.
You can walk around and enjoy the pond, trees, and the restaurant located inside the compound. If you're interested in the vegetarian tofu cuisine of Kyoto you can try the restaurant inside the compound. I know I said I'm trying new things on this trip.... but half my family is vegetarian so I already know I don't like vegetarian food! =P I didn't try it.