The car used for the lessons is a Plymouth Belvedere.The Plymouth Belvedere was a range of cars introduced in 1954. Plymouth had first used the Belvedere name for the top trim level of the Plymouth Cranbrook. The Plymouth Belvedere was offered in sedan, convertible, and wagon body styles, and was produced in seven generations, ending in 1979.
June tells Wally to wait until he's 17 1/2 to get his license. Coincidentally, Tony Dow was 17 1/2 when this episode first aired.
Props used in the show may or may not help determine what state Mayfield was located in, though they could also have been used just for their convenience. In the episode where Wally gets his driver's license, he is seen studying a Vehicle Code Summary booklet from California, of the style used during the governorship of Edmund G. Brown, a.k.a. Pat Brown, in office from 1959-1967.