Hi titank,
I like to go free and easy in Taiwan. My longest holiday record is 8 days in Taiwan, but I only manage to comb thru Kaoshiung and some parts of Tainan. So with my slow coach way of travelling, I really cannot advise how long it will take to cover whole Taiwan.
For example in 7 days Taichung/Taipei, there is Sun Moon Lake, cingjing, Taipei zoo, 小人国 to visit. Then in Dec there is the flower festival (花海)in Xinshe, Natural Science Muesum. These are the only places that I can visit, there are many more places ( Puli 弹珠气水厂, xinshe castle, lavender garden , Miaoli strawberry picking, wuqi fishery port, flying cow ranch etc) that I have missed.
For 8 days in Kaoshiung, there is cijing island, 佛光山, 龙虎塔, 旗山老街, guanziling mud spa。 Then in Dec there is purple butterfly migration in maolin, ciaotou flower festival. I have to skip oyster farm visit, tomato havest picking, Tainan 老街, 国信爷 temple, 爱河,打狗铁道故事馆, 渔人码头,爱河。
Within the 8 days trip to Kaoshiung, and my 7 days trip to Taichung, I can only cover 50% of what I want to see. Very very slow right.....
The only theme park that I visited is 小人国 in Taipei, quite fun and it is not crowded on a weekday visit. I can take as much photos as I want, no photo bomb...haha
I would like to recommend National Palace Muesum in Taipei, you get to see real treasures on display. Generalissimo Chiang Kai-shek brought the treasures from the forbidden palace when he flees to Taiwan.
As for farmstay minsu, there are some in nantou , on the way to cingjing, You can browse thru the following website to know more.
http://cingjing.com.tw/cjcjha/hotelbest/hotelbest.asp
There is another favorite farm minsu that singaporeans like to visit, The flying cow ranch.
http://www.flyingcow.com.tw/
Hope the info above help in your planning.