It's pretty amazing how easy it is to get around with a combination of the mass rapid transit (MRT) system and high speed rail (HSR).
When I was last there, you still needed to take a bus or taxi from TPE to the city center. Now right from TPE airport, you can be in either Taipei city center in 30 minutes (MRT) or on the other end of the island in short order (HSR). Which is kind of fun because the southern end of the island has a totally different vibe.
Within Taipei, you can get to where you need to using the MRT and walk (fun because there's so many interesting nooks and back alleys anyways) or use bike rentals. There's also abundant buses as well connecting from the MRT stations.
Taichung has a single MRT line that connects east to west right off of the HSR line.
Kaohsiung has a few MRT lines and surface light rail that connect the major attractions.
All of it quite cheap to get around, incredibly clean and well maintained, and just all around made it much more enjoyable wandering the city.
The only time we felt the need for a car was visiting the more remote north/east coasts and only because we wanted to save time versus waiting for the slower trains that run those routes.
Whenever I land in SFO, I wonder who's bright idea it was to not have an easy connector to CalTrain at Milbrae (or I'm doing something wrong -- I feel like I don't fully understand the right way to get to Milbrae from SFO). Then on weekends, there's barely any service so if you land on a Sunday for a business trip, you'll waste so much time waiting for the train.
Indeed, it is jarring when we returned to the US.