Ryoanji Temple is one of the most famous rock gardens in Japan. It’s a fairly interesting place (really Japanese) that has some interesting architecture. It’s something that can be done fairly quickly (less than an hour), but I would imagine (I’ve only been in summer and early autumn) that it is best going in nice weather – it is an outside rock garden! It’s a bit quiet and peaceful. This is within walking distance of Kinkakuji, so most people combine them – they’re around a 20 minute walk.
Ryoanji is open from 8:00 to 5:00 from March 1st to November 30th and 8:30 to 4:30 outside of then (so December 1st to the end of February). It is 500 yen for adults and 300 yen for elementary and middle school students.
I recommend walking from Kinkakuji; should you choose to take separate them, you can get to Ryoanji from JR Kyoto station by the number 50 bus. When coming from Shijo Kawaramachi by the number 15 bus.