Before I actually start on providing information about cost of living, it is important to understand Taiwanese currency. In Taiwan, widely used currency is called New Taiwan Dollar, or in short, NTD.
New Taiwan Dollar is a stable currency that can be exchanged in most Asian countries, with an average rate of 28,5 to 31,5 NTD to 1 USD. Just keep it in mind that NTD reached it record high in the past 20 years (Jan 2011,) so things might look a bit more expensive at the moment (comparing to USD.)
The good website that I found for checking on exchange rate is xe.com. On this site you can practically look up and check mid market exchange rates for any currency and look up historical data to give you an idea at cost at specific moment.
For simplicity, I will use standardized exchange rate of USD 1 = NTD 30.
The link to the website that points you directly to NTD is here:
To my surprise, one of the good spot to actually exchange your country currency to NTD is at the airport in Taiwan. Some banks in Taiwan will not want to exchange your money, so if you decide to do it inside the country, just make sure you go to a larger bank to avoid problems.