Subway
https://gyazo.com/55b270ac032d9664e645ba4fa7994ded
https://gyazo.com/4c9e5da41ce0d00f67d470ec31a685af
The Karasuma line runs north to south through the center of the city along Karasuma street.
It's a very convenient and central train line
It has connections to almost all the other major train lines.
https://gyazo.com/5ed1467bc3d20fc3dd760c1159d90fd5
The Tozai line (literally East West line) runs east to west.
Notes
There are many other trains that run underground and throughout Kyoto, but these two are officially called the subway.
Also see Using trains for a map and other general information