You can get hostel rooms in Taiwan, but for cheap accommodation also consider bed & breakfasts (B&Bs). They can be less expensive than hotels in Taipei.
You will sometimes feel tiny earthquakes in Taiwan, but most of them are not a big deal.
Visit the outer islands in the off-season (November is great) for fewer crowds and better value-for-money.
Rates jump for hostels, hotels and other accommodations in Taiwan during Chinese New Year and on Saturday nights throughout the year.