Taipei is nothing but a shopping district. A Japanese we met in China described it as a big night market and found out to be true. If you really wanted to experience Taiwan, go outside Taipei. There are too many places to visit. We were supposed to go to Taroko but instead stayed in Hualien.
There is no train directly from Taipei Airport to the City center so take the bus 1841 (83NTD, 50 minutes). We stayed in Ximen Duckstay Hostel for 2 nights and it’s one of the best hostel (awarded) in the city (cleanliness, price, activities like free music jam) however, its location is not that good. For one, it’s hard to find it from the MRT. Use maps.me offline map, it’s really useful.
From Taipei, we opted to go South. Hualien faces the Pacific Ocean on the east with the Central Mountains on the west. It offers natural sceneries perfect for biking and hiking. Take the Taroko Express from Taipei for 440NTD / $14 (2 hours) but if you plan for just a day trip, take a return ticket for a much cheaper price (780NTD actual vs 880NTD budget).
On the train station is an Information Center, few steps from the train tracks. It’s best to stop over, ask for a free map and advice. They’re friendly and can speak good English. Taiwanese people speak better English than those from Mainland but they have a stronger accent easily discernible.
Food and accommodation are higher priced than Thailand and China. I was expecting it to be cheap. We hang around convenience stores and street stalls but still, it costs more. Someone told me (unverified), Mainland China is putting pressures on Taiwan so cost of living is high.
In case, you plan to travel further to Taroko, you can book a cab from the Information Center or hire bikes. It’s just 25 kms away.