The bad news if you're traveling from Taipei to Hualien is that the High-Speed Rail (HSR) doesn't have a line (yet) along the East Coast of Taiwan. The good news is that the TRA has fast, comfortable, clean, modern trains that connect Taiwan's capital with Hualien City for as little as 340NT$.
It's also possible to get to Hualien by bus, taxi, plane, car, and scooter, and I will give you all the details you need on this page.
The easiest (and most comfortable) way to get to Hualien from Taipei is by normal train from the TRA. You can get on a train at Taipei Main Station 台北車站. You can easily reach that station by MRT (metro). Two to four trains leave every hour.
The fastest trains you can take are the Puyuma and Taroko. Travel time on these two trains is very fast - they cover the 170km in just 2 hours! Ticket price for these trains is 440N$.
The slower trains are the Tze-Chiang Limited Express and Chu-Kuang Express. The journey on these two slower trains take between 2.5 and 4 hours. A ticket will cost you 440NT$ on the Tze-Chiang, and 340NT$ on the slower, older Chu-Kuang.
You can check train schedule and fare here.
If you can buy your ticket in advance, you should definitely do so, especially if you'll be traveling on the Puyuma or Taroko trains, as they tend to fill up quickly, especially on weekends and holidays. That being said, I never book my train ticket in advance and I always manage to get a seat. Maybe I'm just really lucky. Or maybe that's because I usually travel during weekdays. On Chinese New Year, you simply won't be able to have a seat on a train if you haven't booked ahead. You must reserve if you want to travel during that busy period.
Mandarin Airlines has two flights a day between Taipei Songshan Airport (TSA) and Hualien: 7:10am and 19:25pm. The flight is short, only 40 minutes. Plane ticket price is around 1500NT$ one way.
To be honest, I find that it's a waste of both time and money to fly to Hualien when you consider the amount of time required to check-in, go through security, board the plane, disembark... It's much more convenient to take the train.
With the convenience and affordability of train travel, there is simply no reason that I can think of why anyone would want to travel from Taipei to Hualien by bus. As far as I know, there's no direct bus which links these two cities. If for some reason you must take the bus (maybe you suffer from trainophobia), then you'll want to first go to Luodong on the east coast. From there, you can take a bus all the way to Hualien.
If you must travel between Taipei and Hualien at night, when trains are not running, you can always take a taxi. A taxi ride will set you back about 3000NT$ and take between 3 and 4 hours. Set the price with the taxi driver beforehand. By the meter will cost you more than 3000$.
The most convenient way to drive from Taipei to Hualien is to take Freeway #5 to Su-Ao (approx. 1 hour), then switch to Road #9 (approx. 2.5 hours). Road #9 is in the mountains and is mostly two lanes wide - one for each direction. It's not a fun drive. Truck drivers overtake anywhere at full-speed, even in blind curves and tunnels. My only advice is to drive slowly and keep a safe distance from other vehicles. Good luck. Drive safely!
The most pleasant road-trip from Taipei to Hualien starts in Xindian where you get on Road #9. This country road takes you through stunning mountain scenery all the way to Pinglin. From there, you ride to Jiaoxi, near Yilan. Then it's all the way down the East Coast on one of the most dangerous roads (still #9) on the island. Same advice as driving a car applies here: keep a safe distance from other cars and trucks and ride slowly!