Near the Kyoto Imperial Palace is Shimogawa Jinja (Their English page is here). It’s a World Heritage site and is fairly famous, but to be honest I wouldn’t consider it a must-see in Kyoto.
It’s very, very close to Demachiyanagi station (a 5 minute walk) and is free. It’s one group of buildings set in a wooded area – it would be entirely understandable if you walked past it without actually realizing it was there. Honestly, it was just a series of buildings and I didn’t think it was anything really special here. It was okay and would make a reasonable side trip if you also went to the Imperial Palace, but I wouldn’t necessarily go there just for it.
There is the Aoi Matsuri (one of the three big festivals in Kyoto) and it appears to have some part in that.
Shimogawa Jinja is free, and is open from 6:30 to 5:00. It is actually really easy to reach from Demachiyanagi station on the Keihan line, and can also be reached by buses 4 and 205 from Kyoto Station (get off at Shimogawa Jinja Mae). It can also be reached by any of the many buses that go to Kawaramachi Imadegawa, but this requires about a 10 or so minute walk.