Winterizing your Scooter
I'm going on suggestions found throughout the thread as this is something I've not done personally. I'm semi-ridiculous and rode mine through the winter. The general consensus is the following:
(Optional) Some folks suggest changing the oil before storage, others suggest after storage. You're going to want fresh oil in the Spring and oil needs to be changed once used regardless of mileage because it breaks down. There's strong opinions on either side or the argument - just be aware that it should be addressed before or after storage.
1) Fill up the Gas tank with fresh fuel.
2) Treat with
Sea Foam or
Stabil or some other fuel stabilizer (Sea Foam usually gets the most recommendations).
3) Run the engine for 5-10 minutes with the treated gas.
4) With the engine off, drain the float bowl on the carb.

Be sure to re-tighten the drain screw
5) Remove the Battery and connect to a Trickle Charger (Battery Tender Jr. comes highly recommended).
6) If you have chrome on your bike, find a good metal polish (like Mother's Metal Polish) or silicone spray lubricant. The idea is to create a barrier between the metal and the air so there's not a rusted hunk where that shiny bit should be.
7) Give it a good cleaning, you're not going to be spending much time with it for a few months. Apply a light wax or protectorate. Some people like Lemon Pledge... but some people like jumping off of bridges - to each his own.
8)Stuff a rag in the Exhaust to keep out any winter nesters. Be sure to remove this when you go to start it up again.
9) Write a list of all the steps that you've taken so you know the exact state you left the bike in. Tuck it in the Pet carrier or tape it to the speedometer.

Be sure to note, if nothing else that you've got a rag tucked in your exhaust!
10a) If parked in a garage, throw a tarp, painter's plastic or a proper motorcycle cover over it and wait till spring.
10b) If no garage is available, find a good spot to secure it to with a decent lock and cover with a weather resistant motorcycle cover. Try not to store in a place that's prone to winds and make sure it's on solid level ground. Use the center stand. If the cover has eyelets, loop a lock or a bungee through it to cinch the cover on the scooter good and tight: a loose cover flapping in the wind will rub up against the paint and if excessive, there's a small chance it'll scuff the paint.
Walk away and wallow in self pitty

for a little while then get on with your winter. Don't start the bike until you're going to be riding it regularly, unless you want to drain the float bowl again, as well as the other steps.
That should just about cover it.