Renting a motorbike in Thailand is a very popular method of transport for those visiting the Kingdom. It is a wonderful way of exploring and saving money on transport that, depending on where you are in the country, can be wildly overpriced for tourists. However, since Thailand routinely ranks in the top 3 of "most dangerous roads in the world," people should think very carefully about whether or not they want to take the risk.
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Driving on the islands can be considered much safer than on the roads on the mainland because especially on the island I am currently sitting on, the roads will only have motorbikes on them since the roads are not wide enough for cars. I have only seen a few cars here at all and to be honest I have no idea how they got them here since there are not car ferries. That is beside the point. I want to look at if you should rent a bike here or not.
number 1: Have you ever driven a bike before?
Thailand, even on the slow moving islands, is NOT a good place to learn how to ride a motorbike. The motorbikes themselves sometimes have very questionable upkeep and the safety equipment might not be all there. For example: The bike I am currently renting has a VERY WEAK hand brake and there are many hills here. The footbrake is fine but I also have 14 years of motorbike experience. Someone with less experience might be heading down a hill and out of instinct reach for the hand-brake only to have nowhere near enough force to make the bike stop and then crash.
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number 2: can you afford taxis?
The island that you are on should dictate how high the taxi prices are. Smaller islands like Koh Chang, Koh Phayam, and Koh Jum tend to have very inexpensive taxis (like less than $3) no matter where you are going. Therefore unless you plan on zipping all over the place and exploring it might be a good idea to simply not rent a bike.
Most of the bungalow operations have good restaurants on site as well as the ability to visit nearby restaurants and bars that share the same beach. Therefore, it might not be necessary to actually travel anywhere if you are happy with your "resort."
number 3: you can always rent one later
If it is your first time to wherever you are staying (and it likely is) you always have the option of choosing to rent a bike later. every single bungalow operation in existance has someone they can call to bring you a motorbike and provided you are renting for multiple days, the cost will be the same anyway. Therefore, you do not need to make this decision as soon as you arrive at the pier. By the time you have ridden on your motorbike taxi to your bungalows, you will have a good idea of what sort of shape the roads are in and can likely make a better decision about if you feel you have what it takes to drive on the roads.
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TL;DR
Over the past decade of being involved in tourism in Thailand I have seen many customers' vacations ruined by crashing a motorbike. This isn't just because of the road rash and injuries and expensive hospital bills, but also because the contract that you must sign stipulates a rather huge payout that you will have to pay if you crash the bike. This is in many cases not even a rip-off. The bike takes time to repair if someone crashes it and not only will the renter be responsible for these expenses but they are also responsible for the lost income of the owner not being able to rent the bike (make sense?)
The islands are a MUCH safer place to drive a motorbike because of the relative absence of commercial vehicles, but the roads also tend to be in worse shape.
Normally it is better to assess the situation in the safety of a taxi then after arriving at your bungalow operation assess if you actually need to get around as much as you think you do.
Stay safe: Saving money and having the ability to explore is a big plus of renting a bike. However, nothing can ruin a vacation quite as quickly as an injury and a motorbike injury, even if it is a minor one, is likely going to result in road rash which takes weeks to heal and although not at all life-threatening, is very annoying when trying to swim :)