Battery - The only accurate way to detect a weak battery is with professional equipment. If battery caps are removable, check fluid level monthly. Avoid contact with corrosive deposits and battery acid. Wear eye protection and rubber gloves. The trunk and floor boards should be inspected for small holes.
Exhaust fumes can be deadly.
Tires - Worn tires will be of little use in winter weather. Check tire pressure once a month. Rotate as recommended. Don't forget your spare, and be sure the jack is in good condition.