The best way to see many parts of Taiwan is by rented motorcycle, preferably a 125cc Vespa-type machine that's easy to ride (no clutch, no gear changes) and easy to park whenever you spot something of interest.
In this Taipei Times article, it's reported that the ministries of transport and economic affairs are working together to rid some of Taiwan's minor islands of petrol-burning motorcycles and replace them with electric scooters. The latter are, of course, much better for the environment and far quieter. I think it's a great idea and hope it happens. Green Island reportedly has 6,000 motorcycles registered for a population of around 2,700, because so many scooter-rental businesses serve incoming tourists.