Taiwan
Taiwan is not a large island - at its longest point at about 400 km, at the widest point less than 150 km. This makes it easy to plan a visit to most places on the island during 2-3 weeks. The island was discovered in 1517 by the Portuguese, hence its former name Formoza - Ilha Formosa, which means a beautiful island. They already knew then that it was worth visiting!
You will be happy to hear there are plenty of tours out there to take you around the sights and sounds of Taipei often missed by tourists. There is a lot to see and do in Taipei, which is why you should stay longer in Taipei than any other city in Taiwan. You are probably going to start and end in Taipei, so you want to stay in the capital for at least a couple of days.
If you are traveling to Taipei for the shopping, you are probably going to be spending lots of time here. There are plenty of places you can mix up with the locals and get familiar with the old, timeless ways of Taiwans capital. One of the most popular day trips from Taipei takes you to the nostalgic past, where there are small, narrow alleyways lined with stands and ancient traditional tea houses. Tea lovers can embark on a day tour to experience the exquisite flavors of Pouyong tea in the plantations and ancient streets of Pinglin, the area that is at the heart of the centuries-old Taiwanese tea trade.
Full of restaurants, boutique shops, art galleries, and intriguing industrial designs, you could easily spend 5 hours here -- but I would actually recommend heading over to nearby Sinyi district to experience a taste tour through Taiwans intriguing, delicious flavors on Eat Taiwan Eats Food Tours -- no worries, if you do not have time to make it there by the time their 11am start time. A food tour is a great way to whittle down many of Taipeis must-try foods, whilst spending hours with a local who can teach you everything you need to know about tantalizing Taiwans culture and traditions, its history, and recommend you on where to eat the rest of your time. Four full days will give you enough time to see Taipeis main attractions as well as one to two days worth of tours. This itinerary works great for 2-4 weeks (or 3-5 weeks if you prefer exploring things at a more leisurely pace) starting and ending in Taipei, and is arguably better done in clockwise order.
Now, without further ado -- a really nice day tour from Taipei, better explored solo or with a guided tour if you would rather have someone else take care of logistics and transport (more details below). So depending on your interests, choose the best places that you want to see in one day, as per the descriptions that I am going to give below (my favorite ones are marked by stars); then, you can visit the rest on other leisure times, or during other days of the trip, if you are staying longer than 3 days in Taiwan. For best hiking times, that will be in late afternoon, so that you can take advantage of daylight, sunset, and nighttime views of the city. Other great options on a day tour in Tamsui include learning more about Jiufens history as an old mining town by visiting the Jiufen Gold Ore Museum, or if you are more of a nature lover, take the Keelung Hill Hike to get amazing scenic views of Taipei.