I think you can do a round-island trip. Here is the quest for you:
Taipei -> YiLan -> Hualien -> Taitung -> Kenting -> Tainan -> ChiaYi -> Nantou -> Taipei
This one locate at Taoyuan. If you arrive at this one, I can pick you up to where you stay. 40-minute drive from the airport to the downtown. Sorry government haven't built Metro from the city to this airport.
This one is in the city, Metro is 1-min away from the exit of airport, better than TPE. I can still pick you up to where you stay if you want.
If you would like to have an overview of the city.
Taipei 101 also has the best 鼎泰豐 Ding Tai Feng (for Xiaolongbao).
Can go if weather is nice. Very beautiful.
Really nice tea (鐵觀音 Tieguanyin). You can find a restaurant here and drink tea and relax, but it is a bit far from the city (40 mins).
40-minute by MRT from the downtown. Has river-side nightmarkets. Popular for lovers. Beautify sunset. Also a fishman's wharf is in this area. I know a great cafe which I always go when I go there.
Famous food are:
- Oily Bean Curd(Ah Gei) / Stuffed Brown Tofu 淡水阿給
- Iron-hard Preserved Egg / Iron egg 鐵蛋
- Fried Fish Crackers 魚酥
This site http://www.rtaiwanr.com has many sightseeing recommendations, too.
Jiufen is an old town near Taipei
The town has been used as one model of Spirited Away. Famous food is Yam or Taro Balls (Yu Yuan 芋圓)
Houtong Cat Village a famous village full of cats, full of cats XDDD.
Have decent nice restaurants (all kinds) and street food. But good restaurants people will recommand you all try to copied Japanese.
Eat this combo, only Taiwanese people eat this in the world haha:
蛋餅 (egg rolls) + 大冰奶 (iced milk tea), buy from this kind of breakfirst restaurant.
- Clay oven rolls / sesame seed cake (bread) 燒餅
- Fried bread stick / Deep-fried twisted dough sticks 油條
- Soybean milk / soy milk 豆漿
- Rice & peanut milk 米漿
You show this "熱炒店" to everyone everywhere, they will tell you a place to go. But this restaurant is better to go in group like 4-8 people. I can gather people for you if you want.
But you can explore one that near where you stay / where you at.
Famous nightmarket food:
- Stinky Tofu 臭豆腐
- Steam-fried baozi / Pan-fried bun 水煎包
- vermicelli 大腸麵線
- Oyster Omelet 蚵仔煎
- Giant Chicken Fillet 雞排
- Fried Squid 炸花枝 炸魷魚
- Fried chicken (Popcorn chicken) 鹹酥雞
- Pig Blood Cake 豬血糕
- Pepper cakes 胡椒餅
- Taiwanese sausage with sticky rice 大腸包小腸
- Flaky scallion pancake 蔥抓餅
- Gua bao 刈包
- Tomatoes on sticks 冰糖葫蘆
The definitive place to go is 鼎泰豐, also available in Japan. But Taiwan is the origin. You should try ^^.
Tea Shops are everywhere in the City (where I gained my fat...). The famous one is "50嵐" (same as igaiga san's name).
Taiwan is a tropical island means it is very hot but has good fruits to make juices :D
Shaved ice with syrup 刨冰
Tofu Pudding 豆花
A variation of the soft, slippery soybean concoction
aiyu jelly / ice jelly 愛玉
The jelly is made from the seeds of a plant in the fig family. Usually served with lemon juice, sometimes also with shaved ice.
grass jelly / herbal jelly 仙草]
The jelly is made from the seeds of a plant in the mint family. Usually served with milk / honey, sometimes also with shaved ice.
Please see this: http://edition.cnn.com/2015/07/23/travel/40-taiwan-food
If you need to work while here (hopefully not), I have some favorite cafes suitable for work if you need.
If you need, I know there are many in the city, haven't been to anyone though.