Laurie's attitude toward overalls suggests there was a battle of some degree over wardrobe between parent and child. Many parents accompany their children on the first day of kindergarten, but the narrator is denied this privilege, watching him walk away with a next door neighbor. It would be reasonable to infer that this was not necessarily the narrator's choice. Since she is lamenting the loss of her sweet-voiced nursery school tot, it suggests she is not ready to award the independence Laurie has taken—that it wasn't her idea. 

The narrator also describes Laurie as a swaggering character, which suggests the confidence and arrogance of teenage rebellion more than kindergarten excitement. 

The narrator also states that Laurie forgot to stop and wave good-bye, which suggests he has been instructed to do so. If he had been instructed to do so, this is a failure to comply with his mother's reasonable request, which foreshadows the rebellion and insolent behavior to come. 

While Charles's true identity is only revealed at the end of the story, Shirley Jackson gives us some clues in the opening paragraph when she describes the changes Laurie is undergoing. On the day Laurie starts kindergarten, he no longer likes wearing the same clothes: he swaps his "overalls with bibs" for "blue jeans" and a belt. Similarly, Laurie's demeanor also changes significantly: he is no longer a "nursery-school tot" but instead is described by his mother as a "swaggering character" who does not wave to her from the street corner anymore.

These clues are important because they foreshadow the birth of Laurie's alter ego, Charles, and his many bouts of bad behavior. They are subtle enough, however, to go almost unnoticed by the reader, and this makes the story's surprise ending (in which Charles's true identity is revealed) even more effective.