Option 1: Take the Train
This is the most popular and affordable method of getting to Hualien from Taipei. The express trains will get you to Hualien in about 2 hours, while the non-express trains can take 3-4 hours. It is important to book your train tickets ahead of time, especially on weekends and holidays, as the trains do sell out. A one way ticket will cost NT$440 per person.
You can book train tickets online here.
Option 2: Take a Plane
This option can save you time and is easily booked online. The Uniair flights take around 50 minutes and typically cost around NT$850 per person. It should be noted that these flights depart from Taipei Songshan Airport (TSA), NOT Taoyuan International Airport.
You can book plane tickets here.