Start by changing the oil in the machine. The oil drain pipe is either on the back or side of the machine and you just unscrew the end and tip the machine to allow the oil to drain out into an old coffee can or oil drain pan.
Add new oil (usually 5W-30, but check your owner's manual) by inserting a funnel into the hole where the dipstick goes and pouring the fresh oil in. Pause occasionally when you're adding the new oil to insert the dipstick and check how much oil is in the machine. You only want to fill up the "Full" mark on the dipstick (overfilling with oil can cause major damage to your engine).
Drain the old gas from the tank since gas left to sit for months won't work well. Some machines have a drain valve on the bottom of the gas tank, but if your machine doesn't, removing the rubber gas line on the bottom of the tank will allow the gas to drain out. Be careful; the gas will come out fast when your remove the fuel line. When the tank is empty, don't forget to close the drain valve or replace the fuel line, then add fresh gas.
Remove the spark plug. It's usually in plain site either on top of the machine or on the back. Pull the wire off the end and take out the plug itself using a spark plug wrench (there's usually one in a socket wrench set). Alternatively, you could use a crescent wrench or locking pliers on the metal part of the plug to take it out.
With the plug in your hand you can either go to the store and get a new one (exactly the same number since all spark plugs aren't the same), or if you're ambitious enough, clean and regap the plug before replacing it. When replacing the plug be careful not to overtighten it.
Check the belts on your blower to be sure they aren't cracked or frayed. Two-stage snow blowers have two belts, one to turn the wheels and one to drive the auger and throw the snow. You may need to remove a metal or plastic cover or sheath to see the belts. If there are signs of wear you should replace them, but if they aren't cracked, just replace the cover.
Check the auger assembly to see if it's a sealed unit (no lubrication required) or needs to be lubricated. If it does require lubrication, use the oil or grease recommended by the manufacturer.
Check the treads or the air in the tires. Tires can go soft or even flat over the summer, so you want to get them properly inflated. If your machine has treads instead, make sure the tracks are aligned and move freely.
Start it up-and mission accomplished.