Many people (mostly locals) cycle around Taiwan - but due to the ruggedness of the mountains a fold-up is not appropriate for this! Also, the roads are crowded and dangerous, especially on the (scenic) East Coast.
Taking a fold-up on trains and buses, as one of the other posters suggested, and doing small (day) trips here and there might be a better option.

Hello Frank,

Welcome to Taiwan.

Fold-up bikes can be carried on train. It's not a bad idea to travel in Taiwan with that. To get a bike, there are some good bicycle shops here:

Color, In Very downtown, Near MRT NTU Hospital.
1F, NO. 4, Sec. 1, Ren’ai Rd., Zhongzheng Dist., Taipei
02-23959788
色彩 台北市仁愛路一段4號1樓

Da VinCi Bike http://davincibike.idv.tw/
Two shops:
I. 77, Sec 1, Heping W. Rd. Taipei. Near MRT Guting
台北市中正區和平西路一段77號
II. 83, Sec 3, Heping E. Rd. Taipei. Near MRT Liuzhangli
台北市和平東路三段83號

Alan's Mountain Bike http://www.alansmountainbike.com.tw/index.htm
38, Sec. 5, Roosvelt Rd. Taipei. Between MRT Gongguan and Wanglong
台北市羅斯福路五段三十八號

I'm sure that you can find someone speaking English there. (maybe not very fluently, but good enough to understand you)

There are various kinds of fold-up bikes. You can find out A$80~A$400 ones here. I'll suggest you to buy good ones because you are going to take them home, and bicycles here are much cheaper than they are in Australia.

A good fold-up bicycle is also suitable for a island round trip. Some sections are crowded with trucks, but it is still safe to ride bicycle if you are well equipped.

One of the section as what I said above is Su-ao to Hualien. It's pass by a National park and you'll have a great ocean view. You'll see the stunning cliff stretch into Pacific Ocean. But That's the only link between East TWN to north TWN. So all of road traffic go that way. So be careful to ride on that route or take train from Su-ao to Xincheng (The National Park Entrance from Hualien).

If you are not very into the round trip. Maybe East side of the island is the better place for you to spend your holidays. There are two route in the east, coast line or Valley line, from Hualien to Taitung about 300 km each. Both of them run through National Scenic Area. you can do a little round in the east. I think one week is enough or you can take slower pace.

Well, there are so many good places for bicycle trip in Taiwan, especially with fold-up bicycle. I can give you too many advices. Enjoy your trip.

hi frank,
for foldable bikes. look for a shop called "metro oasis" at:
1F, 21, Section 1, Zhongshan Road North, Taipei.
(台北市中山區中山北路一段21號1F)
Tel: 02-25643702
Fax: 02-25417011

it's located near a popular backpackers called "happy family". a short walking distance away from Taipei Main Station.

round-the-island route is now popular among cyclists here. u basically go around (anti-clockwise) the island along the coastal road. from taipei in the north, all the way to kaohsiung in the south, and back to taipei. 10 days in a leisure pace. about 900 km in all.

also u can do side trip to "taroko gorge". this section is a thigh-burner.

hope u find it useful.