Unless you are going for Western food, the fancy restaurants in the big shopping centers or scattered around the city might not be your best bet.
My best experiences with Chinese food included walking around the city and finding some small back street that maybe had an old lady cooking for the locals. You can find these places anywhere, and generally the food is well cooked, so you won’t have to worry.
Recommended things for eating like this: niu rou mian (beef noodles), shui jiao (Chinese dumplings), and different types of xiao cai (finger food). like zhua bing (Chinese-style pancakes). If you're looking for more expensive things to eat, Japanese restaurants in Taipei are really good; in fact, I had a better experience with these than with eating Japanese food in Japan. Be open to try new things. One of my major hobbies in Taipei was to go somewhere unknown and buy something I knew how to pronounce on the menu even though I had no idea what it was. This gave me lots of memorable experiences.