Holders of valid US passports can enter Taiwan without a visa for stays of up to 90 days. US passports holders with less than 90 days of validity can stay in Taiwan until their passport expiration date. If you are coming to Taiwan for tourism, study, or work for longer than 90 days, you should apply for a visa at the Republic of China (ROC) embassy in your country. Applications for visa extensions within Taiwan should be made at the National Immigration Agency.
If you wish to work in Taiwan, your employer will have to get you a work permit; you will also need a resident visa. The Bureau of Consular Affairs has more information. Upon entering Taiwan with a resident visa, you’ll need to apply for an Alien Resident Certificate (ARC) within 15 days of arrival. Details here: link.