Michael Turton, an extremely knowledgeable American who's lived in Taiwan for more than 20 years, has written this very useful introduction to ten of the best cycling routes on the main island. I've been on nine of these roads (and may have been on the tenth - I'm not sure, it would have been about five years ago) and recommend all of them, even if you plan to drive or ride a motorcycle rather than pedal your way around Taiwan.
To mention just three of the ten... Highway 8, which links Hualien with Lishan via Taroko Gorge, involves a huge amount of climbing. Road 193 follows the boundary between the southern plains of Pingtung County and the rugged uplands near Maolin and Sandimen; there are some historical sights, too, such as Wanjin Basilica. Road 159A is an excellent approach to the Alishan area; exploring it one time by motorcycle, I went up a side road which was very promising until I reached the rockslide pictured above.
The wonderful thing about cycling, of course, is it's so easy to stop anywhere you find something of interest - whether it's piles of discarded oyster shells, like these I came across in Dongshi, Chiayi County - a temple or even an abandoned haunted mansion.