Flight
EVA Air has direct flights from NYC (JFK only), Los Angeles, San Francisco, Seattle, Chicago, Houston, Dallas, and D.C. to Taipei.
Flight
China Airlines has direct flights from New York (JFK only), Los Angeles, San Francisco, Seattle, and Phoenix to Taipei.
Flight
United Airlines has direct flights from SFO to Taipei. This can be a good option to consider especially if you want to collect some United miles. I also found this to be more budget friendly for east coasters because for whatever reason the fare is much lower flying with a layover (NYC-SFO-TPE) vs. flying direct on a Taiwanese airline.
Travel Note
A Taipei necessity. An EasyCard (悠遊卡) is a reloadable contactless smartcard essential for getting around Taipei. It costs a non-refundable NT$100 (~$3 USD) and is used for the Taipei Metro (MRT), buses, and convenience store purchases. You can buy and top it up with cash at any MRT station, 7-Eleven, or FamilyMart. Buy one after you land in Taipei!
Travel Note
There are many ways to go from the airport to the city center. Listing a few options that I personally use. The Taoyuan Airport MRT (often called the Airport Express) connects Taoyuan International Airport (TPE) to central Taipei. It operates two train services, the purple Express Train (35-40 minutes) and the blue Commuter Train (approx. 50 minutes), both bring passengers to Taipei Main Station. Uber is available even though the Yellow Taxi drivers say otherwise. So my only tip for booking Uber is to find the exit that is far from the Yellow Taxi line so they can't harass you. Other rideshare apps I recommend are LINE GO and Yoxi.
Travel Note
Taipei is a very public transportation friendly city. The most reliable option is the MRT (aka subway/metro). Can't go wrong with the rideshare options like Uber. The yellow taxies are everywhere but they might not be the most friendly to foreigners and might take an inefficient route so the fare increases due to the distance. The buses can get complicated so unless you are in for an adventure maybe save it for the next visit.