I'm going to offer you 12 places that should be experiences. It'll range from famous landmarks to places that aren't advertised as much. If I haven't been to a place then it won't be on this list. I'm only mentioning places I've been to and have found to be either fun or interesting. They are in no particular order.
1. Fushimi Inari
This is one of the most famous shrines in Japan. Not only is it beautifully set in the southern mountains of Kyoto but it can also be used as a great form of exercise. If you are an outdoors kind of person then you'll enjoy walking the whole path at Fushimi Inari that goes up the mountain and back down. As you're walking the path, you'll be going under a bunch of gates. Facing forwards the gates look ordinary but once you turn around you'll see Kanji written all over that are names of people who contributed to the shrine. If you're more of just a sightseer then you can just walk a bit of the path like I did and enjoy the shops located at the bottom