Modified from an itinerary I put on Wikitravel a few years ago - you need to tell us your group’s travel preferences and interests to get something less generic.
Morning
Start your day early by eating a traditional northern Chinese breakfast. Yonghe Doujiang (永和豆漿) is a popular 24-hour outlet located throughout the city. Fuhang Doujiang (阜杭豆漿, Shandao Temple station exit 5) is a popular eatery, though the long line on weekends is not worth the wait. Although doujiang (soybean milk), shaobing (sesame flatbread), youtiao (long Chinese donut), and other northern snacks do not originate from Taiwan, these mainland transplants were popularized in the Taipei suburb of Yonghe in the 1950s by refugees of the Chinese civil war who arrived in Taiwan in the late 1940s. The descendants of the 2 million refugees who arrived from Mainland China around 1949 comprise only 13% of Taiwan’s population, but over a third of the population of Taipei where they dominated the government until the early 1990s.
After breakfast take a taxi or the MRT to Longshan Temple (Longshan Temple station exit 1), built in 1738 in traditional Fujianese style. You are now in the heart of what was once the Qing dynasty settlement of Bangka. In contrast to the hipper and glitzier districts further east, the working class residents of Taipei’s oldest neighborhood regularly speak Taiwanese instead of Mandarin. The temple opens at 6am. Go inside the temple and watch the locals pray for fortune and good health. Spend about 30 minutes to absorb the atmosphere and architecture. If you’re feeling particularly religious, the incense is complimentary.
Next, take a taxi or the MRT to the Chiang Kai-shek Memorial Hall (Chiang Kai-shek Memorial Hall station exit 5), which overlooks a central square flanked by the National Concert Hall and the National Theatre. If you arrive early enough, you will see elderly people practicing taichi amid the picturesque stone paths and koi ponds. The Memorial Hall opens at 9am; expect to spend an hour there. Climb the stone steps to see the bronze statue of Chiang Kai-shek, flanked by two motionless guards. A changing of the guard ceremony occurs every hour (but if you plan to visit the Martyrs Shrine in the afternoon, feel free to skip the show for later). Descend the elevators to the ground level to view the exhibits on Chiang’s life, complete with his sedans and uniforms.
Ready for lunch? Exit through Memorial Hall’s northern doors at the ground level and through dazhongmen onto Xinyi Road, and take a taxi or either the 20, 22, or Xinyi New Line buses for a short ride along Xinyi Road over to Yongkang Street, famed for its collection of local eateries. If you’re in a group of two or more, try the original Din Tai Fung (鼎泰豊) restaurant near the corner of Yongkang and Xinyi Street, serving the best xiaolongbao (小籠包, steamed pork dumplings) the world has to offer (arrive around opening at 10am to avoid the wait). Or visit Gao Ji (高記) just around the corner for the same but without the crowds. Tu Hsiao Yueh (度小月) serves Tainan-style danzai noodles. Remember to head to Yong Kang 15 for some shaved ice for dessert.
Afternoon
Take the MRT to Dazhi station and transfer to a taxi or the 208 or 267 buses for a visit to the National Martyrs Shrine, built in classical Ming architecture to hold the memorial tablets of the 330,000 war dead of the Republic of China. A changing of the guard ceremony involving two sets of sentries occurs every hour when the memorial is open from 9am to 5pm. Stay to watch this before heading to your next destination.
Alternatively, if you’re no fan of modern Chinese history and military ceremony, head over to Taipei 101, the eighth tallest skyscraper in the world. From Yongkang Street, take a taxi or the Xinyi New Line or 20 bus along Xinyi Road. The building houses an upscale shopping mall in the first four floors and a food court in the basement. To visit the 89th and 91st floor observatories, head to the fourth floor of the mall to board the world’s second fastest elevators. Expect a line at the bottom and provide yourself with about an hour at the top. After you’re done, take the MRT to Dazhi station. From Dazhi station transfer to a taxi or the brown 13 bus to the National Palace Museum. Given the tourist crowds at Taipei 101, this option is ideal only if you are ready to head there by noon, or if you’re visiting on a Saturday when the National Palace Museum closes two hours later, or if you are prepared to skip the National Palace Museum entirely.
From the Martyrs Shrine, take a taxi to the National Palace Museum, which houses the most complete collection of Chinese art and artifacts in the world. The bulk of the museum’s holdings were once housed in the Forbidden City in Beijing as the most-prized possessions of the Chinese emperor. During the final weeks of the Chinese civil war, these items were shipped to Taiwan under the orders of Chiang Kai-shek to keep them out of Communist hands. The collection is so expansive that only 1% is exhibited at any given time. In the Main Building, start from the 3rd Floor and work your way down, visiting the permanent exhibitions before visiting the special exhibits on the 1st Floor. Try to arrive by 3:30pm, so you will have a good 3 hours before the museum closes at 6:30pm. The museum opens until 9pm on Fridays and Saturdays, when admission is free after 6:30pm to Taiwanese citizens.
Evening
No visit to Taiwan is complete without a visit to a night market. From the Palace Museum, take the 304 or S19 bus to Jiantan MRT station for a visit to the nearby Shilin Night Market for a dinner of traditional Taiwanese snacks. The market has an indoor food court, as well as numerous food carts in the streets between the Jiantan MRT station and the Shilin Temple. Try some oyster omelets (蚵仔煎), ba-wan (肉圓, round pork dumplings), fried chicken steak (炸雞排), fried buns (生煎包), meat soup (肉羹), oyster vermicelli (蚵仔麵線), pearl milk tea (珍珠奶茶), or if you’re adventurous, stinky tofu (臭豆腐). Remember not to get too much of the same thing so you can try different foods from different stalls. If you can’t read the menu, just point to what others are eating. After dinner, head to the streets of the night market to shop for clothes and other goodies.
After your visit to the night market, take the MRT to Ximending (Ximen station exit 6), the heart of Taipei’s Japanese-influenced youth subculture. People-watch the pedestrianized streets, which are particularly lively once high school students are let out in the evening.