Ooh nice! The first 3 days of photoshoot will be pretty tiring so probably the 4th day can have a day of relaxation viewing some beautiful scenery places. Below are some that are highly recommended, you can view their official website for more details and pictures. The list is not comprehensive but I would say the scenery of the nature parks are very beautiful! If you love nature and animals like me then you should check out the national parks

Hope this helps!
Scenic Sites:
1. Taroko National Park, in Hualien
http://www.taroko.gov.tw/English/
2. Kenting National Park, in Pingtung (They have underwater world, animals, pretty beach ocean)
http://www.ktnp.gov.tw/
3. Taipingshan National Forest Recreation Area, in Yilan
http://tps.forest.gov.tw/
4. Yang-Ming Shan National Park, in Taipei
http://www.ymsnp.gov.tw/HTML/ENG/INDEX.ASP
Cutural Sites:
1. The National Palace Museum in Taipei.
http://www.npm.gov.tw/en/home.htm
2. The National Taiwan Democracy Memorial Hall (ex. Chiang Kai Shek Memorial Hall), inc. CKS Cututal Center (National Concert Hall & National Theater) in Taipei
http://www.ntch.edu.tw/
3. The office of President, in Taipei
(
http://www.president.gov.tw/en/)
Life Style:
1. Shulin & Tienmu Night Markets, in Taipei
2. Tamsui & Lukung Old Street
3. Taipei 101 Building, in Taipei