Nothing is quite as exciting as purchasing a new car. Even if it's a "used" new car. Don't get too caught up in the pretty shiny metal and chrome, it's what's under the hood that counts. Take a serious test drive before you make a decision. Used cars bought from a private owner don't have any warranty with them. You don't want to buy someone elses problem.
Do you homework. There is a lot of help available on line when you're purchasing a used car. You'll have the vin or vehicle identification number. With this number you can pull up a vehicle history through your state. Do a search for the model and year of your car online. Were there any specific complaints or call backs on this particular model?
The first place you should test drive this car is to your trusty mechanics shop. With a bit of notice he'll be glad to set a bit of time aside to go over the car with you. He'll take a look at your car with a keen and unprejudiced eye.
Take with you whatever service records are available on the car, these will help your mechanic understand the vehicles past. Your mechanic will check the hoses and belts and condition of the oil. These are easy fixes if they need changing or replacing. Your mechanic will check the radiator fluid for coolant level and to make sure the fluid is going to protect in severe cold conditions. He'll see how much oil, if any is present on the engine. Perhaps the car has been burning oil. He'll check the transmission fluid.
Another thing he'll check is the lights, inside on the dash, and outside to make sure they work. He'll check shocks, breaks and tires and alignment.
He'll take a look at the body. Is there any rust underneath on the frame? Does all the paint match, or has the car been repainted? Is everything in alignment? Do the doors open and shut properly?
Let your mechanic take the car out for a test drive. He may be able to pick up subtle things, perhaps a shutter or hesitation, or hear a sound that you may not. His is the best advice you're going to get in this situation.