with the US passport, no visa needed, but can only stay 30 days.
taiwanese airlines have a funny way to price their ticket. if you go there for 2-3 days, they have whole package including flight and hotel stay for ~HKD2000.
I booked a return ticket with 29 days in taiwan for about $2500 with Eva. The longer you stay in taiwan, the more it cost.
zuji is very good for booking flights in asia, but if just flying to taiwan, you can just go to eva website and see what they have.