I personally enjoy planning out my day by going to an area and just wandering around. This document contains a list of various locations where you can do that, with several food/activities that you can do in the area. I've done many of the sample days listed below.
Taipei points of interest
Taipei 101
This is one of the tallest buildings in the world, and central to many attractions. You can easily spend a day or two here just wandering around.
Sample day: Arrive on the MRT red line to the Taipei 101 station. Go visit the higher floors in Taipei 101 (you can get tickets inside), and get brunch at the food court on the bottom floor. Visit the Sun Yat Sen Memorial Hall, and browse through the many malls in the Xinyi area. Take a hike up Elephant mountain (it gets crowded at sunset!), and return to Taipei 101 for dinner at Shin Yeh (but make reservations!).
National Palace Museum
This is a classic museum with the best collections of historical Chinese antiques. They switch things out very often, so it's always worth a revisit. Depending on your interest, you can spend a whole day here, or visit a few places nearby as well.
Sample day: Get traditional Taiwanese breakfast along the way (e.g. Yonghang Soy Milk), and take the bus to the Palace Museum. Get (or bring) lunch while browsing through the various antique collections. For dinner, take a bus to Shilin Night Market and feast on Taiwanese street food.
Keelung River
The Keelung river runs along the north side of Taipei, and is host to many parks and biking paths.
Sample day: Take the MRT brown line to Zhongshan Junior High School and grab a Youbike. Path towards the Songshan airport viewing spot, where you can wait a bit to see some large planes land right over your head. Bike along until you reach Dajia Riverside Park, and spend a few hours biking along the river. You can pack a picnic, and/or visit the Miramar Ferris Wheel. End your day by biking to Raohe Night Market, and have some tasty pepper buns.
Taipei Main Station
The main station is host to several large underground malls, and is also very close to several interesting areas. One day is probably not enough to see everything.
Sample day: Start lining up early at Fuhang Soy Milk, then take the MRT to Taipei Main Station (or walk/bike, it's close!). Spend some time browsing the underground malls (as well as their claw machines) for souvenirs. Bike to Dihua Street to take a look at some traditional 19th century streets. Yongle Market is also nearby, where you can browse through various fabrics and grab some street food. Optionally, visit the Yanping Riverside Park. Go southwards to visit the most trendy neighborhood of Ximen. Walk up some flights of stairs to visit FUFUCatCafe, where milk tea is included with the price of admission. Grab dinner at any of the restaurants in the area, or get street food at Ximending Night Market nearby. To end your night, find HANKO 60 behind a theater-like door for some very instagrammable Chinese-style cocktails.
Beitou/Yangmingshan
Yangmingshan is known for its great hiking options as well as many hot springs (it's volcanic). Beitou is a nearby city with some restaurants as well as nicer hot springs. Be careful to go on a nice sunny day: any fog or rain can make your trip up the mountain miserable.
Sample day: Take the bus up to Yangmingshan in the morning. Pick a trail to hike depending on how long you want to stay. If you're close to the Lengshuikeng hot springs at the mountain, stop by and soak your feet there for a bit (the foot pool is outside and free). Take the bus back to Beitou, and visit the Taipei Public Library branch there, along with the other things in the area around there. If you didn't go to the hot springs at the mountain, this is a classic cheap hot springs spot to go to. On your way home via the MRT, you'll pass by Shilin Night Market, where you can stop by for a meal.
Guanghua/Syntrend
The Guanghua area is essentially Taipei's digital mecca for all things tech related. Syntrend has a lot of anime-themed things, including a whole mecha store and a ton of capsule machines. There's a couple of interesting things here, but I would only recommend this area if you just like browsing through a lot of random tech. They have a lot of phone accessories and computer components here. I was searching for some specific mechanical keyboards, but the selection there disappointed me. This area has some cool underground stalls selling low-level electronic components (think tons of resistors and breadboards), so it may appeal to some people. The prices don't seem much lower than buying online.
Sample day: Get breakfast at the area south of Guanghua plaza (around Zhongxiao Xinsheng MRT). Go to Guanghua Electronics Plaza and browse the underground electronics, then go to Syntrend and get food at their food court (there's also a maid cafe on one of the floors, if you're into that...). Once you're tired looking at everything drop by Guanghua Digital Plaza and check it out for a bit (this is IMO the least interesting mall in this area, but people have different tastes). For dinner, check out this traditional Taiwanese restaurant a station away.
Day trip locations
Jiufen
This is a nice little tourism town about an hour or two away from Taipei. Even though this town did not inspire Spirited Away, it has a similar aesthetic and has some cool old streets to explore. Go on a sunny day, or you won't be able to see the views of the ocean.
Logistics: There is a bus (1062) that goes all the way there from Taipei. It gets crowded, so get on it at an earlier stop if possible. There's a stop at Zhongxiao Fuxing MRT which should be fine. When coming back, make sure to get to the bus a bit earlier (there are long lines, especially on weekends). They take EasyCard, but it's not included in the monthly AllPass if you got one.
Recommendations: Visit A-Mei Tea House for a very instagrammable afternoon tea. If you like shrines, there's a large one in the middle of the town. You can also walk the non-touristy streets to get a cool view of the town from above (approximately here). These Taro balls were pretty popular. In general, just wander around and take cool pictures of the lanterns. I didn't do this, but there are also several hiking spots nearby that are an option for the more nature-inclined.
Tamsui District
I didn't enjoy this area as much, but you can take the MRT all the way to the north end and walk along the waterfront. This is another area that depends on good weather, as you want the good views across the river. Historically Europeans built several forts and residences in this area, which you can visit. There's also a cute free restored Japanese-style residence as well.
There's a food court in the historic area with good coffee. Of course, there's also plenty of street food along the way. If you have time, try taking the ferry across to Bali Old Street and explore there (I haven't gone yet, no suggestions there.)
Taipei Zoo/Maokong
I would strongly recommend spending a day in this area. The Taipei Zoo is easily accessible at one end of the brown line MRT. They have a pretty large variety of animals with a lot of space, as well as a very popular panda exhibit. You could spend a whole day here if you didn't want to rush, but there's something even better right nearby: the Maokong gondola and Maokong area.
Take the gondola for a few dollars and you’ll be able to see great views of Taipei while you go up to Maokong. There are several hiking trails at the top - pick one (or more) that fits your schedule and climb around. After your hike, drop by 猫空茶屋 for some tea or ice cream.