do you mean an electric kick scooter or a motor scooter?
every year the Taiwanese walk a pilgrimage to honor the water god that they like there, Mazu, so my instinct is for you to follow that route.
everywhere is connected by roads in Taiwan with nature trails all around, so you'll be pretty safe to start off from wherever you land, but some roads will be busy or so I assume that the mazu pilgrimage route will be a little more off the beaten path, although I know it goes through some towns and cities.
and all the cities are near the beach so you can go from the beach and then 1 mile in the mountain starts climbing up, so you'll be able to see everything very easily on an electric scooter.
ya know what, let's see.
it specifically says it's a foot pilgrimage, but I can't imagine anyone minding you on a scooter, especially if you're not going during the actual pilgrimage month, which is usually April.
that site has all the information and the route.
Taipei is probably good to start in just because their roads are going to be so good, and you'll want to at least see Taipei first, but you're not going to have any trouble getting around Taiwan.
i prefer hualien as a city, it's a lot slower but it has everything and is a very comfortable town to live in.
I love Taiwan, you're going to have a great time. their breakfast foods are second to none. but then their Buddhist vegetarian buffets are also second to none. their curry is pretty good too. and their fried mushrooms. oh and their sushi is great
I might see you there, haha.