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.
UPDATE: 2/26/18: The Taiwan Railway Administration (TRA) announced in February 2018 that it will be operating two additional Puyuma Express train services each day to and from Hualien County starting March 2018, to promote tourism. The TRA is adding two additional Taipei-Hualien direct service trains each day from March 1 to June 29 that will give priority to tourists traveling to Hualien County. According to the TRA, visitors will have the option of booking their train tickets while booking their hotels.The Hualien Commercial Hotel Union, which encompasses all the registered hotels, motels and hostels in the county, will get tickets from the TRA 20 days before the scheduled train departures and they can sell those tickets to tourists who have made hotel reservations in the area. Remaining seats that are not sold will be open to the general public for purchasing, without a hotel reservation, 12 days before the scheduled train departures. This would guarantee that all tourists with a reservation have a seat on the Puyuma express train on their intended date of travel, the TRA said. The two train services leave Taipei City at 8:57 a.m. each day, arriving in Hualien County at 11:08 a.m., and departs Hualien at 5:30 p.m., returning to Taipei at 7:41 p.m. The TRA has estimated that this will make room for 80,000 additional seats on trains between Taipei and Hualien.
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.