Hi, based on the places you would like to go, this is the possible sequence
[Changi - Taoyuan airport]
[Hsinchu]
leofoo village theme park - never go before can't comment much. Seems like a out of nowhere place from google map.
[Taichung]
feng jia market
Other popular places include: Miyahara ice cream place, the church inside dong hai university, yizhong shopping street.
[Nantou]
farm stay @ cingjing farm - has horse skill performance, sheeps roaming around awaiting for ppl to feed them food, a 'swiss garden'. Lots of mosquitoes up there. You can spend a night there.
sun moon lake and to take cruise
formosan Aboriginal culture village - this itself is a mini theme park.
After all these, can choose to take the High Speed Rail from Taichung back to Taipei.
[Taipei]
Beitou hot spring - Beitou at the northern part of Taipei has a handful of Onsen places
[Pingxi]
jiu fen, shi fen - These two places can be done in 1 day. You can buy the 1 day pingxi ticket for shi fen and many other stations within the pingxi line. at Shi Fen you can put sky lantern. After finishing pingxi line tour, take train back to RuiFang and proceed to Jiu Fen.
After Jiu Fen, you can choose to take a bus to Keelung which is a harbour small town to settle your dinner and also shop there. From keelung, can take a 1hr train back to Taipei.
[Taoyuan Airport - Changi]
There are many other possible combinations to do such as to fly in from Kaohsiung and fly out from (Taoyuan) Taipei.