From Kyoto-station, using JR Sagano line, get off at Saga-Arashiyama station. It will take about 15 min. No need to change trains.
Arashiyama (嵐山) is a pleasant, touristy district in the western outskirts of Kyoto. The area has been a popular destination since the Heian Period (794-1185), when nobles would enjoy its natural setting. Arashiyama is particularly popular during the cherry blossom and fall color seasons.
The Togetsukyo Bridge is Arashiyama’s well known, central landmark. Many small shops, restaurants and other attractions are found nearby, including Tenryuji Temple, Arashiyama’s famous bamboo groves and pleasure boats that are available for rent on the river.
North of central Arashiyama the atmosphere becomes less touristy and more rural, with several small temples scattered along the base of the wooded mountains. The area north of the Togetsukyo Bridge is also known as Sagano, while the name “Arashiyama” technically just refers to the mountains on the southern bank of the river but is commonly used to name the entire district.
More info: http://www.japan-guide.com/e/e3912.html
Arashiyama monkey park is about 15 min walk from the station where you will get off (Saga-Arashiyama station).
Here is train planning website in English;
Here is Arashiyama Monkey park’s website;