Sargent Carter and Gomer seem to meet for the first time on this episode even though they'd already had some rather memorable exchanges on the pilot for this series, that aired a few months earlier on The Andy Griffith Show.
When Gunnery Sargent Carter reports to the Captain, the Captain actually says "At ease Gunny", not "At ease Captain".
Lt Col van Pelt's hat has a plain, black visor. A Lt Col in the Marines would have the gold braid usually referred to as "scrambled eggs" on his visor.
Sgt Carter is seen wearing his Drill Instructor hat indoors not only here but in several scenes during this part of the series. Proper protocol would have him removing his hat whenever entering a building. This could be a deliberate error by the producers seeking to establish and maintain immediate identification of various characters in a show.
In one of the barracks scenes, Carter moves in close to Pyle, who is standing in line. As the camera closes in for a very tight 2-shot, the shadow of the mic boom clearly moves up the bodies of the characters, and then back out again when the shot widens.
When Sergeant Carter visits the Lt. Colonel to get Gomer thrown out of the Corps, he announces himself as Gunnery Sergeant Carter; however, the Lt. Colonel's response is to say "At ease, Captain".