Food and drink is plentiful, cheap, and tasty in Taiwan. It's very hard to go wrong, but here are the summary and highlights of some categories of food that I tried.
Night Markets
Taipei has several night markets, scattered around the city. They serve popular street food, with some specialties between each market. The most touristy markets are Raohe, Ningxia, and Shilin Night Market, but if you have time there are several other ones that are also very good. In general, you'll need to bring cash and be ready to wait in lines (at least for the good stuff). They're open late, typically 6pm-12am - great for the night owls.
Shilin
This is the big touristy one, with many crowded streets and attractions. Be ready to wander around for several hours. Recommendation: Hometown BBQ Chicken (家乡碳烤鸡排) near Jiantan MRT. There's a cool temple within the market (Shilin Cixian Temple). Shilin is somewhat out of the way for most Taipei locations, but if you're making a day trip to Tamsui or Beitou it's a good place to drop by.
Ningxia
This one is just one long street close (walkable/bikeable) to Taipei Main Station, with several well known oyster omelets that have been Michelin-reviewed. Recommendation: "Yuen Huan Pien Oyster Egg Omelet" (get it to go for no line). Nearby attractions: Taipei underground malls (along the MRT stations), Yongle Market, Dihua Street.
Raohe
This one is on the far east side of the city, near Songshan (MRT Green line). Relatively close to Taipei 101. Recommendation: Black pepper buns at Fuzhou Ancestor Pepper Pie. Nearby attractions: Wufenpu Shopping District (clothing wholesale market), riverside parks.
Taiwanese food
There are several classic variants of Taiwanese food that I think are worth trying.
Breakfast
Fuhang Soy Milk is the restaurant to go to for the quintessential Taiwanese breakfast. They serve fried donut sticks in crepes (shaobing), as well as fresh soy milk. The line will be extremely long - if you don't want to wake up early or wait ~1 hour, then you can go to any other breakfast place and grab a shaobing with some soymilk for ~100 TWD.
Lunch/Dinner
Beef noodles
There are two well-known beef noodle shops in Taipei: Lin Dong Fang Beef Noodle, and Yongkang Beef Noodles. Both are worth trying out, and have slightly different styles.
Braised pork over rice
卤肉饭 (Lu Rou Fan) is a classic Taiwanese dish that you'll see many variations of. My Stove is a nice sit-down place with a great version of this, as well as many other Taiwanese classics that haven't been named here. Jin Feng Braised Pork Rice is a classic hole-in-the-wall shop that's bustling and cheaper, with many other classics.
Gua bao (刈包)
Gua bao is another classic food, consisting of a bun sandwiched with pork belly and other savory ingredients. 台北橋頭魯肉飯 at Yansan Night Market sells these as well as their own variant of braised pork over rice.
Nice restaurants
If you're entertaining your parents/some other important guests/just want to have a nice time for reasonable value, you have several options for some classic Taiwanese food. Shin Yeh is a Taiwanese chain that does this very well: Shin Yeh Shiao Ju is a bit out of the way but is a nice relaxed location for dinner. Shin Yeh Mitsukoshi is located right next to Taipei 101, and has a more banquet-like feel. I would strongly recommend making reservations beforehand; if you show up at the door, it can easily be a 2 hour wait.
You may have heard of Din Tai Fung before; it's a famous dim sum chain that has locations even in the US. I would recommend visiting at least one of their (many locations) in Taipei. The waits are long, but there's several locations within malls and you can wander around while waiting.
Dessert
The traditional taiwanese desserts can be found in many night markets. Specifically, you should look for dessert soups with tofu/aiyu jelly, as well as shaved ice and mochi (刨冰/雪花冰). These are great on a hot humid summer day, and like everything else, not particularly expensive.
Yansan Night Market, a very local grouping of street stalls, has a great shaved ice shop at 魏祥記純糖麻糬. Linjiang Night Market also has a good shaved ice location, but I don't remember the specific name.
Boba
Taiwan is the birthplace of boba, and there is no shortage anywhere. Beyond the frequently-seen boba shops lining every night market, you should visit the place that claims to have invented boba: 春水堂 (Chun Shui Tang). Their original location is in Taichung, but there is a convenient branch location in Xinyi, right next to Taipei 101. To be honest, I didn't feel that the boba in Taiwan was particularly better than the solid chains in the Bay Area (just cheaper).
Alcohol
Taiwan basically just has one well known distillery: Kavalan. They specialize in whiskey, but have also been selling gin recently as well. You can visit their distillery in Yilan for a tour if you're interested. You can buy their whiskey at pretty much any alcohol store, and even in some convenience stores as well: I found a good gin and tonic drink by Kavalan in a random 7-11.
Otherwise, there are several good bars to visit with unique characteristics. HANKO 60 has a very cool theme of a hidden theater. It's located in the Ximen district, which has several other bars to visit.
Snacks
Pineapple cake The typical tourism souvenier to bring home from Taiwan is a box of pineapple cakes. Sunny Hills is a nice place to buy these prepackaged and ready for gifting. They also have a cool quirk of serving you a sample of pineapple cake along with some tea, free of charge. According to my memory, Taiwanese pineapples are a different variety compared to the pineapples that we have in the US, so the pineapple in the cakes are sweeter and more fragarant.