On the slopes of Yangmingshan in the north of Taipei City, Beitou can easily be reached by taking the MRT. Famous for its hot springs, it is a great place to relax and appreciate some Taiwanese history.
The town has a hot springs museum and also the Ketagalan Culture Center (凱達格蘭文化館), which is a center for Aboriginal culture in Taiwan. The source of the hot springs, Hell Valley (地熱谷), is just a short walk away and the steamy pool has an other worldly feel. If you fancy a bit of a hike you can visit Liuhuangku (硫磺谷) or the Sulphur Valley.
The Taiwan Folk Arts Museum (北投文物館) is located up the hill and is well worth a visit to appreciate the past influence of Japanese culture in Taiwan. The museum is located at No. 32 Youya Road in Beitou (北投區幽雅路32號). It is open from 10:00 to 21:00 and closed on Mondays. Admission is NT$200. It has a restaurant, but I didn't check the menu. It is quite a walk up the hill to the museum so you might consider taking a bus or taxi.
The Puji Temple (普濟寺) is part of the Shingon sect of Japanese Vajrayana Buddhism in Taiwan. It was originally built in 1905 using donations from Japanese railway employees. It is dedicated to Guan Yin (aka Avalokiteshvara). The temple is only small but beautiful and very well maintained.
Another notable landmark in Beitou is the Beitou Public Library. The library is built from wood and is one of the most notable examples of green building design in Taiwan.
If you want to take a bath in the hot springs Beitou has lots to choose from. The outdoor public baths are just NT$40. Check the opening times displayed at the entrance as they close for cleaning once every few hours. If you have more money to spend there are plenty of places that offer more luxurious and private hot springs.