There are a few things you can do to save you a lot of money.
Clean out your gutters
Disconnect and drain outside hoses
Insulate walls and attics
Weather-strip doors and windows - this will save you on heating costs
Repair roof leaks
Remove branches that could get weighted down with ice and heavy snow
Chimney fires are common in the winter, but they can be easily prevented. Because heat rises, you need to preheat your chimney before you place wood inside. Not doing this can cause "backpuffing" which brings smoke inside the house.
The type of wood you use can also cause smoke to build-up in the chimney. Younger wood, considered green, tends to cause more build-up.
If you're using a space heater, the Consumer Product Safety Commission has some safety guidelines:
Keep flammable material at least three feet away from the heater
Make sure the plug fits snugly into the outlet - if not, use a different outlet
Never run the electrical cord under rugs or carpet - they can overheat and burn
Don't leave the heater on while you're away or sleeping