2 weeks in Taiwan, what other city worth staying?

I've been to Japan and Korea multiple times in the past, but this will be the first time in Taiwan. It will likely be my family so 3-4 people total on this trip some time in September. I was mainly wondering what other city is worth staying in other than Taipei, and if we do what would that city be? I've heard Tainan, Taichung, and Kaohsiung come up a lot. We'll likely exclusively walk and use public transit and I've heard some of the cities outside of Taipei aren't really tourist/walking friendly as Taipei is, so that would be my main concern is if we do spend 3 or so days in another city I'd want the transportation to be very simple.

Depending on what other city we choose to stay in I'd think immediately going there as soon as we arrive makes more sense to minimize moving hotels with luggage and stuff. I'd like to spend the majority of our time in Taipei and use that as the home base. Is that feasible? I've heard you can use the HSR to go to a lot of other parts of Taiwan and come back later in the day to Taipei no problem.