I think the hardest part is the long coach bus ride from from 高雄. It takes 2-3 hours each way... even for me it's a bit too much. Also travelling with a tour may be hard as you have a fixed schedule.
I took my son to Taiwan three times already (visited Taipei, Yi Lan, Tai Chung), but we have relatives there, so the schedule was very flexible and can accomodate our need. My experience is one-hour car ride is quite the max each time. For longer rides, we broke it with lunch and other stops.
If you want sun and beach, I heard that Sanya is a good trying place for first time baby travel. Short flight and short ride.