Here are a couple of tips from a recent visit, though with hundreds of shrines and temples around the city, you won't be short of old stuff to see.
- Walking tour. For our first day we joined the 5-hour walking tour run by WaRaiDo. Mia was our guide. She gave us a great introduction to the city, and we got to see lots of places down the back streets that we wouldn't have seen otherwise.
- Toji (aka Kobo-san) flea market. Held on the 21st of each month, which luckily coincided with our visit. Loads of old and new stuff to see on hundreds of stalls, and lots of tasty snacks too. I went hoping to find a photo album from a Japanese family who'd lived in Hong Kong during WW2. (A long shot I know!) There were wartime photo albums for sale on several stalls, but none with a Hong Kong connection. Market information.
And nothing to do with old Kyoto, but you'll probably be thirsty after all that walking, so I recommend the Craft Man pub, which had friendly service and served a good selection of locally-brewed beers.