Taiwan uses the GSM standard, in both the 900 MHz and 1800 MHz frequencies. If you already own a GSM phone that is enabled for one or both of those frequencies (often called tri-band), you may be able to use your phone in Taiwan through your current provider. For example, T-Mobile has an international roaming plan - but you must call T-Mobile and ask them to enable your account for WorldClass service. T-Mobile rates in Taiwan are $1.49 per minute, incoming or outgoing calls, local or international. People would call your US number and your phone would ring in Taiwan.
A much cheaper option is to buy a pre-paid sim card in Taiwan. You would need to have an unlocked phone that is enabled for GSM 900 or 1800; if you don't have one, you can find a variety of them on eBay. The Motorola V66 usually sells on eBay for under $20; but make sure the phone you get is "unlocked" which enables it to use sim cards from any service provider. Once you get to Taiwan, it is easy to buy a pre-paid sim card for about $10 US; the minute charges are about $.35 US per minute for local calls, somewhat more for international. Incoming calls are free and you will have a local Taiwan number that can be reached from the US. The two main companies that sell pre-paid sim cards are Chunghwa Telecom and Far East Tone.