My Itinerary around the same.
-Reached airport at 6am.
-Take Shuttle Bus To Taoyuan Station.
-Take Highspeed Rail to Taichung HSR Station.
-Go Lavender Cottage + lunch
-Xinshe Castle(Optional)
-Carton King - Taichung(optional)
-Feng Jia NM
-End Day~
Day 2 Then Go to Cing Jing
Sounds good. But ain't you gonna travel around with your backpack or luggage?