We took the Shinkansen from Tokyo to Kyoto, staying in downtown Kyoto. Downtown worked out really well as Kyoto is a big city but spread out with lower buildings so while downtown is a busy area it is not incredibly busy.
Our first destination the next day was the bamboo grove in Arashiyama. We took the train from Kyoto-Kawaramachi, changed trains at Sai , and took the train into Arashiyama Station. The train cars on this route were old, very clean, on time (of course) and just charming. There are a variety of train routes which go into Arashiyama and all of them put you close to where you want to be. The whole area has a lot of attractions, street food and is pretty touristy but definitely worth spending a day. Tenryū-ji near the grove is definitely worth a visit.