You don't need to travel a long way to escape from Taipei's crowds and traffic. There are a number of places within about an hour from Taipei where you can leave the city behind and experience some relative peace. A small warning though: on the weekends many Taipei residents have the same idea of escaping from Taipei and some of these places can be quite crowded. Don't let this deter you though. They still beat hanging out in a shopping mall!
There are a few other popular places that I haven't included here such as Yangmingshan National Park (陽明山國家公園) and Danshui (淡水). In the area around Ruifang (瑞芳) you can visit Jiufen (九份) and take a ride on the Pingxi (平溪) railway. Of course there are other places to visit as well. Don't be afraid to explore!