Scooters are a great way to see Taiwan and offer access in ways that you can't get with any other vehicle. There is an obvious benefit to a scooter in that you can drive it practically anywhere and it doesn't have schedules and service hours like the bus. It's far easier to park than a car which often takes time to find a space and some money for spots in a city. Scooters can filter through traffic, so rush hours become less of a hassle as you slowly crawl your way through the stopped cars and get to your destination long before the person still stuck behind the wheel of his BMW. On the surface, buying a scooter is a great idea.
This goes double in cities which have limited public transit infrastructure like Taichung, Tainan, or anywhere on the east coast. Getting a motorcycle is a great way to get around, and while a bike is FAR cheaper and nearly just as good over short distances, a scooter can open up a lot of areas outside a city. It can also save you money over public transit. All this explains why there are so many of these machines all over the island.
There are a few major drawbacks.
The first is safety. With city roads packed full of trucks cars, scooters, bikers, and walkers there is a really good chance you'll end up in an accident. Unfortunately scooter and motorcycle riders tend to get the most hurt during accidents as your body goes flying through the air. Broken bones, scars from road rash, or even death are not uncommon from accidents involving a scooter. Ask around and listen to stories of accidents, because nearly everyone has a story, and a staggeringly large number know someone who died on a bike.
People in Taiwan don't drive like they do in the west, and sudden lane changes, driving at highway speeds on city streets, violating traffic laws, and a disrespect for other drivers on the road all combine to make scooting a pretty dangerous proposition. If you have no experience driving a scooter - in any country - and have no one with experience IN TAIWAN to teach you, I'd recommend you don't get one.
If you can find someone, local or foreign, with practical local experience to teach you the REAL rules of the road in Taiwan, then you might consider getting one and taking it easy as you learn from them.
Just a few examples of rules you'll need: restricted lanes, speed limits, how passing and signalling is actually done, how to position yourself on a road to stay as safe as possible, two-stage left turns, no turns on red, and where to park in cities which are constantly removing parking spaces to force public transit use.
Roads in Taiwan are repaved constantly so they are always slick with oil when it rains. The white lines - which are everywhere - are slippery and people have been known to low-side on them when it's wet. The roads, despite being repaved constantly, can be incredibly bumpy and occasionally lead to safety issues. In Tainan the scooter lane is seemingly always riddled with manhole covers and pot holes, forcing you to choose between driving in the car lane or taking some good bounces as you ride.
Whenever it rains you will get wet. Yes, they sell nice rain coats for scooter drivers, but you'll need to get some rain shoes if you're tall enough that the poncho doesn't completely cover your feet. Buying a motorcycle? Tag on a pair of rain pants to your purchase of shoes and a coat, and also some method of carrying all that gear around day after day. Scooters have useful trunks built into them which is likely the reason they're so popular here.
Another major issue is getting a drivers license and insurance. If you live in Taiwan, you'll have to change an IDL to a local license within three months getting an ARC. If you want to rent a scooter anywhere, you'll be S.O.L if you are trying to use an IDL, so it's worth it. Basic insurance is cheap and DMV's are surrounded by companies which offer protection, but it takes time to write the written test and schedule the driving segment for your scooter license.
Scooters are also prohibitively expensive if you're only going to be in Taiwan for a short time, as in, less than two years. New scooters and motorcycles run between 30,000NT and 100,000NT. You'll also end up paying registration and road use fees, as well as maintenance and gas costs. While you get a lot of distance for your ~200NT fill up, you'll be spending 250NT + for quality oil and basics at a shop every 1,000KM. Obviously anything else you need repaired adds costs with a per-part range starting at 100NT to around 1,600NT. Repairs and oil changes can be done DIY style - I do all the oil and basics myself - but you'll also need to invest in tools.
I had run a basic calculation of my motorcycle costs and my transportation costs to see when it would be cheaper to have a motorcycle, and it took me over a year to get into the saving money zone. I also bought a used motorcycle off a friend for 20,000 and was driving across the entire city daily. New scooters + reasonably short commutes mean you'd want to take public transit.
You'll run into the police at some point and they'll be eager to see your paper work, especially if they're giving you a ticket. Fines for failure to produce documents will run you 6,000NT and up. They'll also be happy to give you a field sobriety test every chance they get, since apparently there is a belief that white people are more likely to be driving drunk.
Wondering what the legal limit is for drinking and driving? In practice, you should consider the number to be 0.00 on every scale. The police smell your face REALLY close to see if you've got even a hint of alcohol, then off to the breath machine you go. The laws change and get tougher on a regular basis - especially after a rash of DWI-related deaths - and if you've had even a single can of beer in the last two hours, you might be walking home and paying upwards of 90,000NT to get your bike back. Or in jail. The system here is not the same as the BAC limit used in the USA, but the number is around a US .01 or .02 for a ticket right now. The main point is do not drink and drive.
The best advice is that if you live in a city and want to get around on your own is that you should use a bike that you pedal with your feet and wear a helmet. Probably even a half-head scooter helmet. Bargain basement bikes with reasonable parts for rides of any distance and a viably long useful life will run you 7,000NT or so. The cheapest bikes can be had for around 1,500NT, although they're REALLY bad if you will ride a lot. Also, don't drink and bike because it's officially a punishable offense in Taiwan.
If you're going to be going long distances in a city other than Taipei, or live outside a city center, a scooter is a reasonable choice if you cannot afford a car. Be sure you practice, practice, practice, and be alert when you drive. Safety should really be first in your mind as you settle into driving.
To be honest, scooting can be incredibly fun and take you places you couldn't get to otherwise. Rarely do people go on their daily scooter commute thinking about all of the issues above. Yet fewer and fewer foreign folks are scooting regularly as public transit options become better, or they've managed to get jobs where they can afford a car.
The decision to get a scooter is obviously yours alone, but this article is designed to help you weight the pros and cons of buying one. My conclusion is that unless it's absolutely necessary or will save you tons of time (or money) it's best to skip the scooter and find another way to get there.