Actually, getting around Kyoto is a bit of a mess. There is a subway, but it doesn’t run often, is expensive, and you can’t go many places.
The bus system in Kyoto is fairly good, and there is a 500 yen one day pass available, so most people simply purchase this instead. The link for the Kyoto City Bus and Subway site is here. I’ve tried to include Google Maps and relevant transport information where possible, and you get directions from there. If you’re not familiar with getting directions from Google Maps, information can be found here.
You can also rent a bike and cycle around, but in most places bike parking is illegal and the weather might not always cooperate.
Taxis are also possible, but quite obviously they get expensive quickly.
For general transit information (trains only, I believe), I recommend HyperDia. The site is pretty good, but the input is a bit clunky. Also, there’s no way to get the directions or station names displayed in Japanese (so you can print them out and show them to people if you get lost).