Hi Genman,
I've been living and biking in Taiwan for the last 7 years. The East Coast is a really nice place to bike, bacause the cities and towns there are a lot less densely populated. From the airport, you can ride North through the mountains to the Southern end of Taipei, then on through the area near the Taipei City Zoo, and on to the beach at Fulong. Follow the coast to Ilan. I suggest you skip the coastal road from Su-ao to Hualian. There are some beautiful cliffs there, but the entire road is simply a narrow, windy highway for gravel trucks and tour busses. There's little to no shoulder, with a cliff on one side and a mountain on the other... not to mention to treachorous tunnels. If you have extra time, take the mountain road from Ilan to the upslope section of Taroko Gorge, and then just coast down to Hualian from about the 2,600 meter level. If you want to take the train to Hualian, it should be easy, since you have a breakaway bike. All you need to do is pack it up, buy a ticket, and get on the train. Just don't look at anybody like you have a question as to whether or not you can take your bike on. I've only been refused entrance onto a local train with my bike once, and that was in Taidong, where police have a bad attitude towards foreigners.