After agreeing to let June give his old electric train set to neighbor boy Johnny Battson, Beaver decides to pretend the trains are broken and keep them for himself; but Beaver's sneaky plan doesn't account for big brother Wally falling under the spell of little Johnny's pretty, teen-aged sister!
Spooked by a cops-and-robbers movie and absent parents, Beaver calls the police when Lumpy Rutherford comes to the Cleaver house in a "suspicious car" and a gangster costume to pick Wally up for a masquerade party.
Ward finds out first hand why his youngest son's middle name is trouble just by giving his permission for the Beaver to buy an expensive leather jacket just like the one he admired on his friend Richard.
Beaver convinces his parents to let him go in with friends Richard and Gilbert to buy a cute, little burro; but Beaver's promise that "Pepe" will never have to stay in the Cleaver's yard may be impossible to keep.
All the guys are jealous when Eddie Haskell's dad allows him to drop out of high school, especially when he brags about the money he's making in his new job; but best friend Wally isn't sure that Eddie is as happy as he seems to be with his newfound independence.
Wally comes to the rescue after Beaver lets his pal Richard do the Rickover family laundry in the Cleaver washing machine, "bailing" the inexperienced younger boys out when laundry day turns into soapy mayhem.
Beaver is happy to free Wally up for a date by offering to babysit for the Murdock's five-year-old boy, Chuckie, until he finds out that he'll be sitting with Chuckie's ten-year-old sister, Patty, instead.
When Ward urges Beaver to follow in Wally's footsteps and become the next Cleaver basketball star, Beaver tries out for the City Park basketball team. But Beaver finds he lacks his older brother's talent for the sport and, afraid to disappoint his dad, doesn't tell anyone when he is cut on the first day of practice.
Angry with his dad for taking his car keys away, Lumpy Rutherford decides to secretly join the Merchant Marine Corp and has the enlistment information sent to the Cleaver's address instead of his own. Trying to be a good friend, Wally hides the Corp literature in his room; but, when June finds it while cleaning, she's sure that a break-up with his girlfriend and a recent reprimand from Ward have made Wally unhappy enough to leave home.
Wally and Beaver reluctantly give up their Saturday morning to clean the yard, miss the garbage man's pick-up deadline and go from the frying pan into the fire when they trust Eddie Haskell and Lumpy Rutherford to take the trash to the dump in Lumpy's car.
Beaver and his friends buy fad "monster" sweatshirts and agree to wear them to school on the same day but Beaver is the only one who manages to sneak out of his house in the gruesome attire and suffers the consequences of violating the school's dress code...and his parents trust.
When their week-end dating plans are ruined by Wally's promise to take Beaver and his buddies camping, Lumpy and Eddie connive to free up their pal by trying to scare the young campers into coming home early.
While Ward and June are out for the day, Beaver is convinced by his visiting friends, Gilbert and Alan, to split the cost of a long distance phone call to Los Angeles Dodgers pitcher, Don Drysdale. But the prank costs much more than they expect when the boys are left on hold while the famous baseball player finishes his shower.
Ward decides to teach Wally a real-life lesson in economics by helping him and Beaver invest one hundred dollars in the stock market. But when the nice, solid utility stock recommended by their dad doesn't show much activity, the boys decide to take Eddie Haskell's advice and buy risky, but rising, space tech stock.
Inspired to become a reporter by a visiting foreign news correspondent, Beaver runs to the local newspaper after school to apply for the open delivery boy position. But the job goes to a boy who pretends to be sick to get out of school early and Beaver, unwilling to "rat" on the dishonest kid, hopes that the new carrier will make so many mistakes that the paper will fire him.
When he is punished for staying out too late on a school night, Beaver writes a letter to popular advice columnist "Ella" hoping that a sympathetic response from her will convince his parents that he is being treated unfairly.
To teach his youngest son the importance of a budget, Ward lets Beaver join a record club; but the real lesson in financial responsibility comes after Wally's warnings to return the weekly selection refusal cards are ignored and Beaver winds up with more music...and a bigger bill... than his allowance allows.
Beaver takes it personally when schoolmate Shirley makes fun of his hair; but when his efforts to tame his unruly locks don't go as planned, Beaver decides instead to follow troublemaker Eddie Haskell's advice to give Shirley a taste of her own medicine.
Beaver encourages pretty classmate Betsy Carter's crush on him to get her to help him write his autobiography for a school assignment; but when Betsy finds out that Beaver has been calling her names behind her back, her assistance turns to sabotage.