I just had a short transit stay in Taipei. No visa needed for US citizens. The airport train is cheap and efficient. Also, most people are polite and food at mom-and-pop eateries is dirt cheap. Overall, it's a safe and clean city. There are awesome cafes and tea shops. Great for 2-5 days.
I flew Starlux Airlines, which is a new Taiwanese airline. The service level was a full echelon higher than that at US airlines. The food on board was delicious. Highly recommended.