Drain the gas tank. The easiest way to do this is by running the moped or scooter until it is empty. This method also empties the carburetor. Spray some WD-40 into the gas tank. This will help prevent rust. Some people may choose to instead fill their tank all the way to the top, add a fuel stabilizer, and sealing off the gas cap to oxygen-proof the tank.
Change the oil. This will help ensure that any debris is flushed out for the winter. Check the tires on the scooter or moped. Make sure they are inflated to the proper pressure for that tire and scooter. Putting a 2 by 4 under the tires will also help keep them off the cold, hard ground during the winter months.
Remove the battery and store it indoors. If the battery is old, just dispose of it. Charge it once a month with a battery tender. Make sure it is stored in a safe place where it won't get bumped or exposed to flames.