we are looking to travel from Taoyuan International to Hualien , what is the quickest way?

The best way is: first take the bus No.1819 at the airport and get off at Taipei station East Gate 3. the ticket fare is NT$125 and the duration is about 55 minutes. Then take TRA to Hualien. The train ride may take about 2 - 3 hours and NT% 283 - 440 due to different train class and types.

After Spring Festival, there may be the Metro from the airport to the Taipei station and you can pay more attention to that.

I am currently staying in a "home-stay" in Zhuangwei township, Yilan county. It's off season so I got a room for $50 American per night on arrival, and the price dropped to $30 per night after two nights. The place is like a B&B except no host. It backs right up to the beach, and I got it all to myself. The home-stay arranged my airport pickup and shuttled me to the place for $30 American. It was about an hour and a half drive, and the last thing I wanted to do upon arrival was drag my luggage to a bus or train station. Maybe I've paid to much, but I'm grateful for the conveniences. The place I'm at is called The Loha Family Seaside Homestay, The owner Mr. Chen, is very accommodating (pardon the pun) and Qua Qua the driver was also very pleasant and helpful. The best app for communicating to or in Asia is WeChat and you can iron out details with locals that way prior to arrival. I hope that insight is helpful.