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.
In their effort to keep Mayberry quaint and slightly behind the times the production designers have Opie and the kids in town riding conventional Western Flyer style bicycles, even though the ultra modern and ultra popular Stingray style was already in full force around the country.