This is a new experience for me. In all my years, I can count only three real trips by train. Two in Canada when I was a high school student, and one a year ago through China to Tibet.
In Taiwan, all the main cities are connected by train. One could probably go around the whole island in a bit more than a day, all on the train. That was my route, just stretched into 3 weeks.
They are cheap. A 4-hour ride will cost about $20 CAD. They run exactly on time. They are comfortable. You can pay by credit card.
There are different rates depending on the time of day, and one essential app for the traveler's phone is called "Taiwan Railway", adless train schedule, by DIN Lab. The only thing you can't do (yet) is purchase tickets, although they say that's coming.
What I would do is find the departure time I wanted and just show that screen to the ticket agent. Although most of the agents spoke some English, it seemed less confusing to use the phone app.
Trains in Taiwan - More comfortable than a bus and, in my limited experience, better views.