One way I like to take a break from my work online with my business www.littlewonders.com.tw is to browse some of the forums based in Taiwan. Recently, people have been discussing the article published by the Lonely Planet online, ranking Taiwan as the 9th-best country visit in 2012, because of cycling.
I remember my first proper cycling experience in Taiwan. My teammates and I were preparing for a triathalon, and we cycled from Taipei to Wulai, had a barbecue and a swim, and cycled back to the city. At that time, there were really no cycling paths that let you travel a decent distance on the bike, so you just had to grit your teeth and take on the traffic. I would not have recommended family cycling trips at that time! There has been much work put into developing Taiwan's cycling trails over the past few years, and there are now plenty of places to go cycling by yourself or with your family.
As a follow-up to the Lonely Planet article, another Taiwan-based blogger, Taiwanxifu, has taken time to introduce 10 spots ideal for cycling with your family.
You don't need to buy a bike to enjoy the trails. You can rent bicycles from the starting points of most trails. However, Taiwan is "famous" for it's bicycle manufacturing industry, and you pick up some nice bikes at reasonable prices around the island. One thing that is a little harder to find is a decent child-seat for the bike, and if you are moving to Taiwan and have one already, I would suggest packing that up and bringing it over. Other expats are sure to want to buy it off you when you finish with it. (You can sell it at the Forumosa, or Taiwaneasia forums' Sale sections.)