This is actually a fairly simple process that can be made complicated by incompetent bureaucrats. You need to purchase insurance before you register your bike. Registration is not necessary for scooters, but everything above that will require registration. You will be running between 3-4 buildings depending on the layout of your local offices.
There are many auto insurance companies in Korea. So far, I have only had Hyundai Insurance and they are your typical insurance company who will try their best to increase profits over coverage. You can also check with Meritz, Samsung, and many others available as smaller companies. From what I understand, most insurance companies are the same when it comes to insuring your bike and the main difference is price. I haven't purchased insurance online. I went to the insurance office in person and purchased insurance. Just to give you an idea of pricing, the insurance for both my Exiv 125 and Comet GT250R were both 300,000 Won for the year. You can cancel your insurance early and receive a refund and as you probably won't be riding in the winter, this is good way to save yourself a few bucks. The insurance for my Yamaha R1 was just over 400,000 Won for the year. I only had liability and did not bother with full coverage. For the most part, full coverage doesn't give you much extra when it comes to motorcycle insurance as there is no theft protection.
After you have purchased insurance, you will have to go to your city hall or town hall and register your vehicle. You will need your passport and foreigner registration card. They will ask you for a "도장", but most foreigners don't have one so just explain that you'll just give your signature. The word for registration is "등록". Registration requires some paperwork and some fees. They will ask you the value of your motorcycle, tell them 100,000 Won and if they ask you why it was so cheap, tell them you bought it from a friend. There will likely be a bank there, but if there isn't be prepared to find a nearby bank to purchase registration stamps. They will more than likely help you out with all of this so don't worry if you're a bit confused right now. After you have registered, you will have to go back to your insurance company or fax them the information. Now that you're registered, you have to pick up your license plate. There is usually a separate facility that holds all the license plates. You will have to bring your receipt of registration to them and they will bolt your license plate to your bike for 10,000-15,000 Won, this is including the price of the plate itself.
Once you have a license plate, you're all set and ready to ride legitimately. I know it may be a pain to get around to all of these offices by foot or taxi so just take your bike, just ride really carefully. The police won't bother you too much if you ride without registration and if they do, just tell them you're on your way to get your bike registered.
