These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
That sequence was largely filmed with primitive CGI. Since this was The Eighties, the technology couldn't be used to render anything remotely realistic, so they simply nixed the realism part in order to pioneer a new filming technique.
Blue and Orange Morality: The probe. It's willing to sacrifice an entire planet just to make contact with an alien species, and yet it (or its designers) see nothing wrong in this. Could be a case of badly-programmed AI or AI without empathy, but even so, it makes no sense to us.
It's possible the probe would just keep repeating its message until it got an answer, that in effect it could not be turned off until it did. It's still poor planning, as what would happen if there never was an answer hadn't been considered by the programmers.
Fanon: It's suggested by official sources, though not confirmed, that the Enterprise-A was not a brand new ship but was the ship Yorktown (mentioned early in the film as suffering from the probe's effects) and was rechristened to be the Enterprise. In any case it is question of how quickly they were able to make a new USS Enterprise when the first was destroyed only about 3 months prior in story. This story did have the backing of Word of God, as the Yorktown was the original name of the starship in Gene Roddenberry's first pitch of Star Trek. Such a concept would be used in a later series, where the USS Sao Paulo is redubbed the USS Defiant.
An alternate theory is that the Enterprise-A is a new ship, originally ordered with a different registry, but then that wouldn't explain why Starfleet would be so keen to retire the ship less than a decade later. The Yorktown theory would fit, as it was around at the time of the original series.
Sulu's line "San Francisco, I was born there..." has gotten more hilarious since George Takei came out of the closet. Originally it was funny just because of the fact Takei was born in San Francisco, but the revelation adds even more humor to:
Sulu: I love this town!
It also set off a number of jokes about just how Sulu convinced the helicopter pilot to loan him the vehicle - to say nothing of the way he takes Chekov's hand helping him onto the Klingon ship after he gets rescued.
Catherine Hicks, who plays Dr. Gillian Taylor, went on to play Mrs. Camden on the TV series 7th Heaven. Her on screen husband, Stephen Collins, played Commander Willard Decker in Star Trek The Motion Picture.
In the beginning of the film, the Klingon Ambassador says "There shall be no peace, as long as Kirk lives!" The plot of the sixth movie is about a new Klingon-Federation peace treaty, and Kirk is instrumental in getting it ratified and preventing another war.
Scotty, after overcoming his initial uncertainty over how to use a Mac Plus: "The keyboard? How quaint!" Back then, ironic as mice and GUIs were cutting edge. Now, "just using the keyboard" is quaint. With the increasing popularity of touchscreen-based devices, doubly so.
Insane Troll Logic: The Klingon ambassador's tortuous case for how Kirk was supposedly a terrorist trying to use Genesis as a weapon/secret base to destroy the Klingon Empire would appear to be this, depending on whether or not you consider it outright propaganda or at least extreme wishful thinking on the part of the Klingons in order to save face.
Fridge Brilliance: This is the exact same rationale Kruge gave for breaking protocol, crossing the Neutral Zone, and firing upon a Federation vessel in the first place. He did so completely on his own initiative (even telling Torg "tell this to no one"), but it becomes brilliance when you consider that the Klingons figured out in hindsight WHY Kruge would have gone to Genesis, and latched on to that as the only defensible explanation. Not having all the facts hurt them (the Ambassador specifically says the Genesis device was test-detonated by Kirk himself, which is a double falsehood), but with Genesis' existence already causing galactic controversy, protecting their interests makes sense for them.
Never Live It Down: Good luck finding anyone who knows much more about Jacqueline Susanne and Harold Robbins than their being mocked as "the giants" of 1980s literature in this film.
Nightmare Fuel: Though pretty lighthearted compared to previous entries this comedic entry does have some very creepy scenes, such as the Federation ships being de-powered by the Probe, and the strange montage that plays out in Kirk's head when they're going forward in time.
One-Scene Wonder: The punk on the bus that Spock gives a Vulcan neck pinch to. Quote the actor, who was also the Associate Producer of the film: "I could win the Nobel Peace Prize and my grave would still say 'Punk On Bus – Star Trek IV'."
Strawman Has a Point: The pompous Klingon ambassador demands justice in response to Kirk killing Kruge's crew in the last movie, which cues Sarek to explain just how villainous they really were. The Federation president assures everyone present that Kirk will face Federation justice, which the Klingon ambassador scoffs at. Out of all the things the crew did in the last movie, no one is going to bat an eye over what happened to Kruge's crew. However, the Klingon ambassador nonetheless winds up being right to scoff at "Federation justice." All charges of theft and sabotage of Starfleet property are dropped, and Kirk's violating the chain of command is "punished" by a demotion to Captain and the command of his own ship - the two things he wanted anyway.
Visual Effects of Awesome: The whales. It got to the point where US fishing authorities were brought in and criticized the producers for being around them. All the while not realizing they were just small scale animatronic models.