During the towns welcome to the governor, the town band plays a tune that should sound familiar. It is the same tune later used as the theme song for Mayberry R.F.D.
Andy says the museum even has Martha Washington's lavalier. A lavalier or lavaliere or lavalliere is an item of jewellery consisting of a pendant, sometimes with one stone, pendulous and centered from a necklace.
John tells Andy "Jesse Earle Hagen is coming over from Mount Pilot to play bass." Earle Hagen was the composer for the music of The Andy Griffith Show. He was also the whistler heard in the theme music on The Andy Griffith Show (1960).
The "Mayberry RFD" theme that the previous comment refers to is called "Mayberry Theme" and has been used since very early in the first season.
During the town meeting to make plans for the parade, it's suggested that they try to get Sam Eply on his motorcycle to bring up the rear. Apparently they succeeded. When the parade is turning onto Main Street you can briefly make out a man in the distance on a motorcycle at the end of the line.