A lot of people who visit and relocate to Taiwan are curious about getting out on two wheel in what has been described as a cycling paradise. The idea this island is paradise is a misnomer and puts way too much pressure on this island to perform. Taiwan is most correctly called a good place to cycle.
All of Taiwan's cities are easily biked because the infrastructure is set up for scooters. You'll be moving slower than scooters and you won't be treated the same way, but there is definitely more respect than you'd get in most western cities. Drivers in Taiwan rarely have issues with bikes, and they're not driving badly to piss you off or to make a statement about the place bikes have in society. If they're driving poorly it's just because they're bad drivers, plus the driving culture here is based on perceived privilege and the idea that might makes right. There are also some rather glaring flaws in the legal system for punishing drivers who hit pedestrians and bicycles making the results of hitting or killing someone much less of an issue than it is back west.
Somewhat unsurprisingly the biggest safety issues tend to be buses. I'm loathe to say anything bad about the bus drivers in Taiwan since many are very professional and provide a necessary service. Yet the majority of them also have some bad habits when it comes to bikes, perhaps because there is no mechanism in place to punish them for doing it. Expect to get cut off or slowly driven into the curb by buses which refuse to yield to cyclists. It happens in every city, and on every country road. When turning they turn quickly without waiting for pedestrians and run red lights, while honking, much more frequently than cars do. Luckily they're huge, and you've now been forewarned. Get as much distance from them as you can while staying safe, and either hang back to let them go first, or beat them by enough distance you won't get cut off at their next stop.
The next biggest issue is cars/small trucks which are turning, with an emphasis on left. Failure to yield right of way on left turns is the single biggest cause of accidents in Taiwan. Cars which turn left rarely look for anything other than oncoming cars or scooters, so pedestrians and bicycles are unlikely to factor into their decisions. Everyone turns quickly, trying to grab that moment of open space to make it across the way. Seen a few walkers and bikers get hit this way, so keep your head up and hands ready for action. When cars are turning right they WILL see you if you're in the road, but they will either act like a bus and muscle you off the road or blast past you to squeal on the breaks. There are some drivers who manage to find it in their hearts to wait as you pass the intersection, but don't expect that to be the norm. On streets with open parking you may even get cut off into a parking space with a parked car in front of you.
When you're biking near parking lot entrances, driveways, store fronts, or any of the other areas where cars and scooters can get off the main area of a street, you need to be careful. Vision is usually obstructed for drivers, and the only way they can get out is to just slowly plow their way into traffic. That means that cars entering the roadway will NOT stop for you, and could even hit you as you pass because they expect you to swerve or stop dead.
No one can estimate your speed on a bike. If you're slow, they'll expect to pass you and you'll just stop or slow down. If you're fast they'll think you're going slow, and cut you off anyway making things that much more dangerous. On the off chance you're going fast AND they notice before they try to overtake you to turn right, chances are high they'll floor it just to pass you and then get stopped by something as they try to turn. The advice here is to bike as fast as you can while maintaining your ability to stop on a dime.
Cars rarely follow the speed limit, and this goes double for large roads on the weekends when things are less congested. Nothing is pleasant about a car or scooter passing you at highway speeds when you're having a bike ride. Municipalities have done a lot to drop speed limits on city streets and non-highway areas, but enforcement mechanisms are weak at best and give the police relatively few tools to issue tickets to violators.
When you're in a crosswalk/bike crossing do NOT expect anyone to yield to you while you cross the road and keep your guard up. LOOK at oncoming and turning traffic before you enter the road and don't expect them to suddenly yield as you approach. Right turns, left turns, or even straight, no one seems to understand the concept of yielding to pedestrians. If you ride out too quickly, someone will definitely hit you. There are tons of ways this could be fixed, but the driving culture here does NOT support pedestrian safety.
Cars don't stop at red lights, and you should never, ever be the first one to get out into the street after the light has turned green. Remember the rule just above that you need to LOOK before you do anything. Intersections are often huge and yellow lights are less than a second long. Everyone will wait and look before they cross the street, so make sure you do it too.
Swerving to avoid obstacles is a necessary skill every biker should have, but it can also get you killed. The roads in Taiwan's cities are crowded and those drivers behind you may not be expecting you to move out of the way of an object. Getting buzzed by scooters or cars is the norm during rush hour, so make sure you look before you move out into traffic. Watching a U biker get tagged by a scooter, then smacking into the back of the bus she was swerving around was a bad sight.
Wear a helmet if you're going to ride with any speed on a sidewalk or in the street during rush hour. If you're going to ride a bike, you should wear a helmet anyway, and there is no excuse for not wearing one if you commute by bike.
If you ride on the sidewalk, ring your bell early, and call out with your voice to people as you get close. The bell is impersonal and it does absolutely nothing to make people move. As a bike you have second priority to pedestrians, and as a foreigner people will judge you harshly for even the perception that you're being rude. Everyone HATES the cyclist who pulls up behind them and rings their bell like crazy, so don't be like that. When people move, you could go the extra mile by smiling or waving to them.
On weekends on Taipei's bike paths you should never expect to go fast. If you ride quickly you will quickly end up almost hitting some kid who can't control their bike, the family dog, or running off the path to avoid them. Danshui, BaLi, and the neihu side of the city get comically crowded on nice weekends. If you want to ride the bike trails at speed get out early on the weekends when no one is around, or get out after dark. If you need some place to just absolutely go all out, hit up the road along the north coast and work your way over the mountains from WanLi or along the river roads from Keelung.
If you drink and bike, the police may stop you, specifically on road near any of the major nightlife areas. There was a huge backlash over drinking and bicycling as the Ubike system took off, and bikes are supposed to be treated the same as scooters as far as the rules go. Remember at the beginning of this writeup, how I said there are some glaring flaws in the legal system for punishing drivers who hurt pedestrians? Well one beer and any form of transit, including bikes, will basically land you in jail with fines of 100K NTD, more if you're in a car. Killing someone on their bike, or as they cross the road while sober seemingly costs 300,000NTD or so in insurance payments to the family of the deceased and you're done with the whole process. Litigation isn't the same as in the west either, leaving little recourse to the victims to get more for their loved ones.
The mountains of Taiwan are beautiful and worth biking in their entirety if you have the legs for it. I cannot recommend it enough. Roads are often narrow, at most one lane with a small shoulder, so you need to protect yourself as you climb and descend. There is a lot more patience in the mountains for passing cyclists however, which means you can enjoy a few extra inches of pavement as you rocket yourself to the top. Many drivers do not understand how to take turns however, and even when there are yellow lines they will go well past the middle of the road around a corner. There are a ton of blind corners on mountain roads and cars will probably not see you coming in the mirrors, if they even look. There are a ton of stray dogs, some of which will attack you. Make sure you protect yourself around turns and keep your speed low enough you can avoid dying on a downhill. Many of the most popular rides have minimal protection if you miss a turn, meaning you hit the guard rail and tumble down a steep slope to your eventual death.
So, to review, Taiwan is a great place to ride bikes. You can ride all year round - although night time is best in summer – and there are bikes everywhere. It's not a paradise, because there are a ton of ways you can still get hurt or even die on your bike. The driving culture is awful and cars here don't understand how to interact with cyclists, just like in the west. You can bike the entire island if you have it in you, and there are a ton of new bicycle paths in every city. You can easily be a bicycle commuter and many cities have bike sharing programs with bomb-proof bikes waiting for you to ride around town.
Keep safe, and have fun. Once you get used to the flow of traffic and areas you ride often which pose safety risks there is nothing to worry about. Get into the mountains, bike along the oceans, bike to the night market, and enjoy this beautiful island on two wheels.