The Taiwan High Speed Rail was a very exciting thing back when I first rode it in 2007, which was also the year it opened. It cut the travel time between the two major cities of Taiwan, Taipei and Kaohsiung, from ~4 1/2 hours to ~1 1/2 hours. It is usually roughly US$50 to travel one-way from one end to the other, so many older people thought and still think it is an unnecessary expense and would rather take a much slower (and cheaper) train or a bus.
Tickets can conveniently be purchased online, at the station, or at a Seven. There is a discount for being an early bird (up to 35% off. They released a limited number of tickets at the cheaper rate), a foreigner (I think it is 15 or 20% off and you have to show or scan your passport when buying), or for buying in bulk. If you buy on the day of and are willing to not be guaranteed a seat, you can get a small discount of roughly NT$50, or ~US$1.67, for an unreserved seat.
If you want a guaranteed seat, you should buy one. However, you should also note that some cars do not have reserved seats. If you are starting from one end of the line, whether Zuoying (Kaohshiung) or Nangang (Taipei), it is almost guaranteed that you can find seating in an unreserved car. The large electronic signs in the station by the ceiling will tell you which cars are not reserved. It is typically cars 10-12 or 9-12.
When you have a flight that departs from the TPE, the high speed rail connects smoothly to the airport. Luggage can be stored in the luggage areas.
The cars are showing their age after a decade and a half. It was very shiny and new a long time ago but now looks more like a clean but older train.
