Taiwan Youth Hostel Association has a network of hostels around Taiwan. A complete list of hostels in English can be found here.

To promote travel by young people the Taiwan Youth Commission has set up a Youth Guesthouse Network. It provides accommodation at government hostels for NT$500 or less on Sunday to Thursday. You need to have a foreign passport and be aged 15-30 to take advantage of the offer. More information can be found here.

There are also many hotels around Taiwan. The area around the railway station in most cities usually has a number of hotels. Prices and standards vary, but it shouldn't be too hard to find a room for less than NT$1,000. These hotels often have signs in Chinese only, so if you can't read Chinese ask someone for directions.

Check out the following budget hostels around Taiwan

Gaoxiong (高雄)

• Kaohsiung 202 - near the Love River
• The Orange House - near the Railway Station
  
• International Friendship House - Zuoying near the HSR Station

Hualian (花蓮)

• Hostelling in Hualien - article in Highway 11 

Taidong (台東)

• Taitung Hostel 
• Taitung Traveler Hostel