There’s two types of trains going between Hualien and Taipei, Tze-Chiang Limited Express and Chu-Kuang Express. When you book the tickets online from the official Taiwan Railways Administration website, look for the Taroko Express (太魯閣號) as it is the fastest way to go from Taipei to Hualien at only 2 hours 10 minutes. On the online train schedule, you won’t see “Taroko Express”, but you will see “Tze Chiang Limited Express” with two little green triangles underneath. You can book your tickets either by train type, or train number so have these information ready when you are booking.
The booking opens 2 weeks before your travel date. I would book the tickets as soon as possible because these tickets are in high demand especially during peak season. On my way back from Hualien to Taipei, the express tickets were sold out so I had to take the Tze Chiang Limited Express which took 3 hours to get back to Taipei.
Once you have booked your ticket online, you can print out the e-ticket. When you get to the railway station, you will need to go to the ticket counter and collect the magnetic ticket as shown in the picture. Once you have your ticket collected, you are good to go !
ps: If you need wifi in Taiwan, check out i-Wifii Mobile Router. Rental is only NTD 100 per day and can be shared among 10 people. Renting it is really easy : book it online –> pay online –> use it –> return it at 7- 11 (which is literally everywhere).