what type of itinerary would you suggest for travelling on the last week of Oct. in Taiwan?

So you are planning to tour around for a week? If so, here is a simple itinerary for your reference:
Two days in Taipei: National Palace Museum, Taipei 101, Ximending, Shilin Night Market……
One day in Taoyuan; Then go to Jiayi and have a visit to Sun-moon Lake and Ali Mountain.
After that, go to Kaohsiung and then Kenting;Hualien can be the succeeding destination.

According to the past weather records, it shall be still warm during that time and cloudy days would be the majority together with several lightly rainy days. Adults are suggested to wear lonng-sleeved shirt, and light jeans. T-shirts can also be taken in case. For infants, jackets and thick pants are necessary.

Thank you!

Is it easy to move around with the places that you have suggested, eg, like taking public transport? Do I need to get those day pass or what kind of public transport pass? How about the data plan?

Appreciate your kind advice. Thank you.

Yes, among these spots, you can have easy access to the public transportation. And I suggest you buy an EASYCARD, which can be used on MRT, buses and some trains. As for the data plan, you need to go to any branch office of telecommunication companies, they will get you the most suitable data plan.