Taiwan ... where we spent 6 lovely car-free days near the beginning of our RTW tour ...
There was some beautiful cycling around Hualien where we based ourselves.
This is a road out to a lake we visited.
This is the cyclepath along the foreshore within Hualien.
This is the cyclepath once we leave Hualien and start cycling southward.
The cyclepath crosses the bridge in the distance in the photo above, and then ends. But it is not needed anymore because the roads going south have a great shoulder for as far as we cycled.
Cycling within Hualien was an experience. At first, I didn't like it because it seemed like utter chaos, but after a couple days I got into the ebb and flow of the traffic. Fortunately, most of the roads were wide enough to handle it all.
Incidentally, if you are looking for a car-free holiday, I would recommend Taiwan. We flew in and caught a shuttle bus to a hotel near the airport. It was not recommended to cycle in the immediate area of the airport, but that's OK because the hotel was great. They quite happily shuttled us to the train station even though we had bicycles, panniers and everything with us.
If we had folding bicycles, I think we could have taken them on the train as carry-on luggage, and we may do that in the future, but because we had full-size bicycles we had to check them. However, that was not a problem. Lots of people checked bicycles and scooters and all sorts. And even though we did not speak Chinese, and the train station staff did not speak much English, we were able to communicate just fine.
The train to Hualien was reasonably comfortable, and upon arrival we had no difficulty collecting our bicycles. There was a form I needed to fill out and it did help that one of the other passengers who was also collecting something from baggage, spoke both Chinese and English. He helped me through the form. The people there were very friendly and helpful.
We found a nice, basic hotel for about $30/night, and made that our headquarters while we cycled and walked in various directions exploring the area.
Next time we go, we might do it more as a point-to-point tour, cycling from one place to another ... and we would bring folding bicycles. That would make getting around just a bit easier.