Many expats in Taipei live in Tienmu, the foreigner enclave of Taipei located in the north part of the city. This is a relaxing and quiet area with lots of western restaurants, international schools, and a more relaxed vibe than downtown. That being said, to travel or live in Taipei looking for a neighborhood that resembles Pleasantville USA kind of defeats the purpose doesn't it?
That's why I recommend Ximen or DaAn areas for more of a local feeling. In these areas you'll find the hustle and bustle of a real Asian city. This means more crowds and noise, but also more culture. You'll find cool temples, little parks, all kinds of cheap Taiwanese food and noodle joints, bars and nightlife, and 24 hour buffets. What you won't get is the chilled out atmosphere of Tienmu, Neihu, or Beitou, all in the north part of the city.
Many expats also live in DaAn, Ximen, and Xingyi, where the Taipei 101 is. But they may be the younger crowd, students, English teachers, and young professionals working in Taiwanese tech companies.
My advice is ask yourself what your purpose is, to soak in the Taipei experience, or the suburban Asia experience. Then choose accordingly