Rasbro and I rode around Taiwan just a week after Warren but starting in Taipei. We rented from a Brit called Jeremy at email@example.com. He is not very responsive and I am not particularly recommending him, but there's not much choice in the city. 125cc/150cc old scooters were 2000TWD for 4 days (or only 2600TWD for one month!), no limit on distance but he requested we get an oil change at 500km which we did (150TWD).
Driving is very easy as long as you understand that on a scooter you must give way to cars in cities. I find Taiwanese drivers to be relatively law abiding ( compared to S.E.Asia, India), so do not regard them in the same way as Warren. On mountain roads we were rarely passed as long as we kept to 50/60kph, then on steep inclines everyone is slow. At one point I had to wave past a Porsche that had politely trailed me for about 10km.
First day's route: out of Taipei on route 9 and 5 to Taroko Gorge entrance (about 200km)
Day 2: Taroko Gorge on route 8, over the 3200m mountain peak at Ren'ai on route 14, to Puli, south on route 21 to Sun Moon lake then continued south on R21/R18 through Yushan national park to Fenqihu, a town on the Alishan mountain railway (300km)
Day 3: R162/R149/R3 north from Fenqihu to Nantou city, then R14 back to Puli, back on R8, but then after going through Ren' ai again took R7 to Yilan (320km)
Day 4: morning ride back to Taipei via the east coast R2 to Fulong then going inland to R5 back into Taipei. (150km)
We had the same concerns about the lack of gas stands, particularly me on the 125cc (150cc was more economical), but with knowledge of their locations and a 130km range it was not a problem.
"If at first you don't succeed, try, try again. Then give up. There's no use in being a damn fool about it."