Tires:
When tires sit for a prolonged period of time they will lose some air. Check the pressure before the first ride. Make sure the wheel nuts are tight and that the tires are in overall good condition. If you see any cracks in the sidewalls or in the spaces between the tread, replace the tires. They're the only thing between you and the road!
Battery:
If your scooter has a battery, check the acid level and the charge. Over time the acid will evaporate and the battery will begin to calcify. It is best to top your battery off with distilled water. Make sure your battery has a good charge before reinstalling it in your scooter.
Lights:
Test all your lights. Headlight (low and high beam), turn signals, tail and brake lights, and any indicators in the dash.
Fluids:
If you changed the oil before you stored the scooter for the winter all you need to check is the oil level. If you did not change it, make sure you change it now. All the dirt and grunge that builds up through normal use will be stuck at the bottom of the motor and will begin to circulate through your transmission and ruin any new oil that you put in. For this reason, it's best to run the scooter for a few minutes (ride it around the neighborhood) before draining the old oil. Vintage Vespa tip: Get an old squeeze ketchup bottle for refilling the gearbox. It has a small hole to squeeze the oil through. You know you have put enough in when the oil comes back out the fill hole.
Control Cables:
Make sure the cables are adjusted and in good working order. Sloppy and frayed cables will not last very long. Replacing a broken cable on the side of the road is much more difficult than at home, so if your cables are feeling suspect, lube and adjust them now.
Brakes:
Make sure your brakes are in working order before you start riding in traffic. Sometimes when a scooter sits for a long period of time the brake drums will get a small amount of rust on the brake surface. This will cause the brakes to lose grip until the rust is worn down, so work the brakes a few times to make sure you can stop reliably.
Nuts & Bolts:
Finally, make sure that all the hardware on the scooter is tight and that nothing is going to come loose and fall off. Check your rims and hubs, rack, and crash bars if you have them.