Family trip for 6

Hi, I'm planning our 1st long distance trip for my whole family. I've 4 daughters, aged 19, 14, 13, 9.
I'm planning to go in June 2015. We are hoping to go for free & easy as we are unsure what we should be covering in Taipei. Would it be cheaper to travel by train or cheaper to book transport?

If we stay in 1 hotel for about 5 days, how far can we cover by foot & train? Do we need to change hotels etc?
It's our 1st trip there.
Which hotel locations would you recommend? We would prefer budget ones etc...

Any advise or recommendations would be greatly appreciated.

Do send them to Peiyunhotmail.com.

Thank you.

Angeline