It's possible to tour Hualien in a day. Plan which day you want to visit Hualien then go to Taipei Main Station to buy return train tickets on the day you arrive. Set off early in the morning coz the travelling time for each way is abt 2.5 to 3hrs and return in the evening.
In order to cover more places in shortest time, you can hire a cabbie to visit the attractions on an agreed price. There are many cabbies waiting outside Hualien Train Station to bring tourists around Hualien. You can check out other travel forums for recommended cabbie as well.
The attractions that most ppl will visit are Taroko, Tunnel of Nine Turns, Swallow Grotto, Eternal Spring Shrine, Cing Shui Cliffs, Bai Yang Bu Dao, Ci Sing Lake etc. There is an ocean park (Farglory Ocean Park) further to the south but I don't think you can cover this in a day trip.
As for the attractions in Taipei are:
Ximending, Danshui/ Fisherman's Wharf/ Lovers' Bridge, Tianmu, Chiang Kai-shek memorial hall (renamed as National Taiwan Democracy memorial hall), Sun Yat-sen Memorial Hall, Marty's Shrine, National Palace Museum, Longshan Temple/ Huaxi St Night Market, Raohe Night Market, Shihlin Night Market, Shida Night Market/ Gongguan Night Market, Yang Ming Shan, Beitou, Wulai, Yeliu, Jinguashi, Chuifen, Keelung/ Miaokou Night Market etc
Shopping: Taipei main station underground mall/ Ho Che Zhan, Wufenpu, Miramar Entertainment, SOGO, Taipei 101, New York New York, Shin Kong Mitsukoshi, Eslite Bookshop etc
Outside Taipei, you can visit Maoli for fruit farms, Sun Moon Lake & Yushan at Nantou, Alishan at Chiayi, Kaohsiung's Cijing Island for seafood and sunset etc. These places definitely require more travelling time.
It depends on your preference, nature or cultural tour or more of food and shopping. When do you arrive at Taiwan? Coz there's a flower festival at YangMingShan till mid of march. Strawberry season is from dec to march/april but this year may end earlier due to global warming.
