When Clay Spencer is driving to the university he's in what looks like a 1955 Ford F100 truck. When he arrives at the school he's driving a 1956 Ford F100. Then when he gets back to the library he's again driving the older model Ford truck.
In the scene where Clay is milking the cow, "Chance," she is first seen without a halter, then with one in the next shot, then again moments later without it.
When Clay rushes from the mailbox into the house with Clayboy's college application reply he enters the front door on location, but comes through the kitchen door on the studio set.
If a cow is giving milk, she must have a calf. You wouldn't breed a cow that is still giving milk.
Reverend Goodson tells Clay that Isaac Walton wrote The Complete Angler "...a century or so ago". It was in fact over 300 years earlier.
When Clay Sr. breaks his leg, there is no blood on his pant leg despite Clay Jr. mentioning that the leg bone had broken through the skin.
When all the children are saying goodnight to each other Walton's style, the visual scene is from outside the house. Even though everyone is saying goodnight to each other, there is bright sunlight on the lawn and blue sky is visible through the trees. The scene was quite obviously filmed during the day and therefore this was hardly the setting for a night scene.
Around 00:08:40, camera's shadow is visible on the truck's angle.
Info on the film lists Wally Cox's character as Reverend "Goodman", but he calls himself Goodson, as does the rest of the cast.
There doesn't seem to be any logic for why Clay posts the name, hours, and Open sign for the library on the side of the building instead of the front facing the road.
When Clay drives to the university, he has a run-in with a campus cop, who tells him where to park. Clay parks the pickup truck and heads for the administrative building. Not only does Clay fail to put any coins in the parking meter, but the campus cop standing there does nothing about it.