While Taipei is an interesting city, you don’t really have enough time to explore it. The great think about Taiwan is the incredible natural beauty of the mountains and east coast and the friendly people.
I think you would get the most out of your trip by flying or taking the train down to Hualien on Thursday the evening you get in. Take a taxi up to Sincheng and spend the night at the Hualianji B&B.
No. 63 Beisanjhan Shunan Village, Sincheng Township, Hualian Co.
The next day catch the bus or taxi up to Tiansiang. Spend the day hiking the trails in the area and spend the night at the Catholic hostel there. You can eat more reasonably priced meals at the large hostel above the Catholic hostel. The head down the canyon on Saturday stopping to swim or hike at places like the Cihmu bridge or the Sedaka (sp?) trail further down. Having your own transportation in the gorge is nice. You should be able to rent scooters or a car in Sincheng.
Then head back to Taipei on Saturday evening and hit the town if you have the energy that night. Others can probably give you a better ideas of where to go depending on your insterests.
On Sunday morning, take the subway down to Longshan temple. Make sure you try the Mango ice across the street. I’d also highly recommend visiting the 228 Museum in the 228 park to learn a little about Taiwan’s history.
I would skip (1) CKS Memorial, (2) the Palace Museum, and (3) Taipei 101.