Sulu says he wouldn't know the danger if it were "2 feet away" from him. It is repeatedly established that the Federation and Starfleet use the metric system, not the traditional (or "standard") system.
At the end (in the re-mastered version), as the Enterprise leaves orbit, the view screen shows the planet from the rear of the ship, between the engine nacelles. In that view, the planet is lighted from the left of the ship, but in the last shot, as the credits appear, the planet is lighted from the right. This would indicate that the ship had reversed 180 degrees and flown past the planet again.
Incorrectly regarded as goof: Lela states she has never seen a starship before. How did she get to the planet? However, Its firmly established that Starships are not the only space vehicles that are used to transport humans to other planets. In fact, Starships are rarely, if ever, shown as being used to transport colonists to a planet.
At about 28:30, Kirk beams aboard the Enterprise, but there is no one left on the ship by this point to beam him aboard. Indeed, a moment later, Kirk is on the bridge of the ship -- and there confirms that no one but he is aboard.
The spores are surely the greatest medical discovery in the Star Trek universe: They restore one to perfect health even to the point of regrowing lost body parts. And once a person has been cured of any malady, the doctor would only need to start an argument with the patient to finish the treatment. The Enterprise should have loaded itself to the gills with spore plants and put every doctor in this series and all future spin-offs out of a job. But we never hear from the spore plants again.
Leila brings Spock, wearing his Starfleet uniform to a place in the open countryside where the spore-generating plants are growing. Moments later, Kirk calls Spock on his communicator; Spock answers while lying in the field with Leila, wearing a green jumpsuit typically worn by all the colonists. Apparently Spock found a jumpsuit lying in the fields and quickly changed clothes before Kirk called.
The goof item below may give away important plot points.
(at around 25 mins) Captain Kirk tosses an Omicron pod plant such that it lands in front of the helm station, where Sulu sits. (at around 28 mins) As Captain Kirk enters the bridge, this wide shot shows no pod plant in front of the helm station. (at around 31 mins) The pod plant reappears in front of the helm station. While the pod plants aim at their victims, they are sessile.