First, due to the government regulation, you need to provide passport and a secondary ID in order to apply for a prepaid card in Taiwan. The secondary ID could be a Visa, Tourist Visa, international student card, entry permit issued by Taiwan National Immigration Agency or an official photo ID of the applicant such as Social Security Card, Identity Card, or Driver's License, etc.
The second step is to select a company to purchase a prepaid card from. Now the question is of which company you should pick. The 3 best known telecommunication companies in Taiwan are Chunghwa Telecom, Taiwan Mobile and Far Eastone. Let’s take a look of the signal reception: Chunghwa telecom was the only company that provided telecommunication service before 1996 in Taiwan, it has the best infrastructure built around the islands, as a result it has the best reception in rural areas or islands in Taiwan. The signal reception in cities among the 3 companies have no difference. You can compare the signal reception on this website: http://freqgis.ncc.gov.tw/pub_new/PublicMap.aspx
Except for the quality of signal reception, phone rate is another important factor of choosing a phone company. I listed 3G airtime rates and internet plans of 3 companies below:
3G airtime rates
3G wireless internet plans
You can choose a package that fits your itinerary based on the information above. Among the 3 telecoms, Taiwan Mobile is the only one that doesn't provide 3G internet only (no calls) service. Basically, apply to all 3 telephone companies, you have to deposit some money to a 3G prepaid card for airtime and if you need 3G internet service, you can subscribe to an unlimited data usage plan and deduct the money from your deposit, directly. You can use your remaining credits for airtime as long as your prepaid card is within the valid period. However, there is an expiry date for prepaid cards which is 180 days from the date of last recharge. To recharge your prepaid card, you can go to any convenience store (Including 7-11, Hi-Life, OK Mart, and Family Mart) or the telecom offices in Taiwan.
Chunghwa Telecom: http://www.cht.com.tw/portal/en/Location
Taiwan Mobile: http://english.taiwanmobile.com/english/customer/serviceLocations.html
Far Eastone (Chinese only): http://www.fetnet.net/cs/Satellite/eCare/ShopCenter
PS: iTaiwan Government Indoor Public Area Free WiFi Access is available to foreign visitors:
you can present your passport or entry permit (for Mainland China residents) at a Travel Service Center located at any major airports, train stations or mass rapid transit (MRT) stations in Taiwan and give the service personnel your e-mail address. After the service personnel applies for an account number for you, you can use the account number to access the Internet at any major indoor public area in Taiwan free of charge.
Or you can simply go to the Taipei-free public Wi-Fi (TPE free wifi) website and register an account with your native mobile phone number. The TPE free wifi account can be used to log onto iTaiwan service. Have fun!