These are tried and true methods for helping get your scooter through the cold winter months in northern climates. Some of the instructions are vintage-centric; feel free to edit them to make them better suited to all scooters.
Battery
If you only do one thing on this list, this should be it.
Scooters use lead-acid batteries. The charge in the battery is what keeps the electrolyte's freezing temperature very low. When it sits for a long time, the battery will slowly lose charge, raising the freezing temperature. Once the acid freezes once, the battery is useless. If it's being used regularly (like in a car) this isn't an issue, but if you put your scooter away in November and forget about it until April, you'll probably have to buy a new battery in the spring.
The best way to store a battery over the winter is to remove it and bring it into a warm place. It's important to charge the battery once every 2-3 weeks. A trickle charger such as a Battery Tender is perfect for this job.
If you must leave the battery in the scooter, charge it once a week, or keep a trickle charger on it all winter if possible. It has a much better chance of surviving the cold this way.
Gas Tank
Over the winter fuel can break down and turn to varnish. This causes things to gum up (including carburetors and fuel valves), and the fuel won't burn as well as fresh gas. Additionally, scooters with steel gas tanks can rust inside over the winter from condensation. There are many different opinions on what to do to prevent these things when you put the bike in storage. All have merit. The most popular two are:
-Fill the tank all the way to the top with gas and add a bit of Sta-Bil, Seafoam, or other fuel stabilizer. This prevents gas from breaking down over the winter and keeps the sides of the gas tank from being exposed to air and moisture.
-Drain the tank completely, and spray the sides of the tank with an oil-based spray such as WD-40 to coat them and protect from moisture.
Carburetor and Motor
Fuel that breaks down can clog your carb jets over the winter. It's a good idea to put some fuel stabilizer like Seafoam in the gas and ride the bike a couple miles before putting it away for the winter. This will keep the gas from turning to varnish. If you have a manual fuel valve, turn it to the "off" position and run the scooter until it dies. Open the choke and start it again, running it until it won't run anymore. This will drain nearly all the gas out of your float bowl.
If you're really adventurous, you can remove the carb and spray it out with carb cleaner, paying particular attention to the main and idle jets. This will make certain there's no possibility of old fuel gumming them up.
If you leave old oil in the gearbox all the dirt and deposits will settle out of the oil and sit in the bottom of the gearbox. If this happens every time you go to change your oil the deposits left in the gearbox will contaminate the new oil as soon as you put it in. So, it is a good idea to change your oil before you put the scooter into long-term storage. This applies to crankcase oil also.
Cylinder and Exhaust
Plug or cover the tailpipe. This will keep moisture from creeping up the pipe and into the cylinder. A wadded up rag or a plastic bag and a rubber band will do the trick.
The cylinder is also a very straightforward prep. Remove the spark plug and pour a capful of motor oil or some "fogging fluid" into the cylinder. This will lube and protect both ends of the crankshaft and cylinder. Now gently kick the motor over once or twice to spread the oil around the crank and cylinder, then replace the spark plug
Other Parts
If you're going to store the scooter outside, make sure you have it covered and well protected from the elements. If you can, get it in a shed or a friend’s garage. Some scooter shops offer winter storage for a reasonable fee. If you'll be storing the scooter long-term (more than just over the winter), it's a good idea to put it up on blocs and deflate the tires to keep the tires from rotting.