Here's an ideal itinerary for 6 days in Taipei:

Day 1: Arrival and Exploring the City

• Arrive in Taipei and check into your hotel
• Head to Taipei 101, one of the tallest buildings in the world, and take in the stunning views of the city from the observation deck
• Visit the nearby Elephant Mountain for a hike and more breathtaking views
• Explore the Xinyi District, known for its high-end shopping and dining options

Day 2: Cultural and Historical Sites

• Start the day at the National Palace Museum, which houses an impressive collection of Chinese art and artifacts
• Visit the Chiang Kai-shek Memorial Hall, a grand monument dedicated to the former president of Taiwan
• Explore the Longshan Temple, one of the oldest and most famous temples in Taipei
• End the day at the Huashan 1914 Creative Park, a former winery turned cultural center with art exhibitions, shops, and restaurants

Day 3: Food and Night Markets

• Start the day with a visit to the Shilin Night Market, one of the largest and most popular night markets in Taipei
• Try some of Taipei's famous street food, such as stinky tofu, oyster omelets, and bubble tea
• Visit the Raohe Street Night Market for more food and shopping options
• End the day with a visit to the Taipei 101 Night Market, located at the base of the iconic tower

Day 4: Day Trip to Jiufen and Shifen

• Take a day trip to the charming mountain town of Jiufen, known for its narrow streets, tea houses, and stunning views of the coast
• Visit the nearby town of Shifen, famous for its sky lanterns and picturesque waterfall
• Return to Taipei in the evening and relax at your hotel

Day 5: Nature and Hot Springs

• Start the day with a visit to Yangmingshan National Park, known for its hot springs, hiking trails, and stunning views of Taipei
• Relax at one of the hot springs in the park, such as the Beitou Hot Springs or the Wulai Hot Springs
• Visit the nearby Beitou Thermal Valley, a geothermal area with hot springs and bubbling mud pools
• End the day with a visit to the Taipei Zoo, one of the largest zoos in Asia

Day 6: Shopping and Departure

• Spend the morning shopping at the Ximending shopping district, known for its trendy fashion boutiques and street performers
• Visit the nearby Red House Theater, a historic building that now houses art exhibitions and performances
• Depart from Taipei and head to the airport for your flight home

I hope this itinerary helps you plan your trip to Taipei! Let me know if you have any questions or need any further assistance.