Mathilda – When we were in Japan last spring, we stayed with someone who had been living there for 3 years. He recommended making advanced reservations for the longer, well-traveled routes, like Tokyo to Kyoto. Which we did. We bought our Japan Rail Pass online before we left – I just looked at the confirmation email and it says we got it through Japan Experience, which is this website: https://www.japan-rail-pass.com/.
As to whether you should buy one – If you are just taking the train from Kyoto to Tokyo, it might not be worth it. There are folks on the Japan forum at Trip Advisor who will look at your itinerary and give advice on whether to buy one. The Japan rail website is pretty user friendly – you can look up your routes, total the cost, and see if it makes sense to buy a pass. We used ours in Tokyo too. The website also tells you what track # your train is leaving from, which is really really helpful.
In Kyoto we took a walking tour of Gion and a biking tour that were both wonderful. We also loved the Fushimi Inari shrine and hiked to the top. I planned most of the trip using trip advisor and japan-guide.com.