The convertible Aunt Bee buys and learns to drive is a mid-1950s Ford Fairlane Sunliner. On some shots, you can see the "Sunliner" badge on the door. (Ford Motor Company was the vehicle supplier for the show.)
The car is a 1955 Ford Fairlane Sunliner (convertible), standard (With a clutch).
When Aunt Bee is taking her first driving lesson, she turns around in the middle of the road causing the tires to squeal loudly. However, the road on which she is driving is an unpaved dirt road. The tires would only squeal on a paved road surface.
Goober is selling his 1955 Ford Fairlane Sunliner for $295. This would be the equivalent of $2,697.57 adjusted for inflation in 2022. The car was 10 years old at the time. In 2022 they run as much as $60,000 to collectors.
Having Aunt Bee wear a scarf-strapped hat when she's driving looks cute and quaint, but it's actually a costuming device to hide the fact that a stuntman is driving the car.