Take the JR line to Arashiyama station, walk over the bridge and up the massive hill (it's worth it). When you get to the top, adorable monkeys will scurry between your legs and take food from your hands.
Walk from Kyotoshiyakushomae station towards the Imperial Palace, and you'll encounter a little but delicious restaurant called Hafuu. After you're done stuffing yourself with beef, nap under the trees in the Imperial Garden, but make sure to watch out for cicada shells falling on your head.
Take the JR line to Inari station, walk through the hundreds of Torii gates at Fushimi Inari Shrine, then stop by the pond to stalk the many species of turtles there, including the particularly strange Chinese softshell turtle: