The best price found from United States to Taiwan is on May 7th, 2015, travelling with EVA Air. See more options, compare prices and book: search flights.
Most people book flights through a local travel agent, online at www.eztravel.com.tw, or through worldwide sites such as www.expedia.com. If flying to the outer islands, book directly with the airline by phone, via their website, or at 7-Eleven ibon kiosks.
Taiwan has two major international airlines. Eva Air started operation in 1991 and has had no fatalities to date. China Airlines was somewhat infamous for its safety record in the 1990s. Incidents since the turn of the century have been few. Officials credit this to new training practices (with pilots training at US flight schools) and a new corporate culture.