I'll start off with some observations and nitpicking
Still a fan favourite and consistently voted the top ST film. Easily the most shown on TV.
When Harve Bennett (producer of "Mod Squad", "Six Million Dollar Man" and "Bionic Woman") was called in by Paramount he says: It was a total surprise to me (They) said, "Can you make a Star Trek II for a television-type budget? For less than $45 Million?" I said, "Where I come from, I could make four or five movies for that."
The entire original cast appeared with the exception of Majel Barrett and Grace Lee Whitney. Kirstie Alley made her screen debut as Saavik. But Bennett did not dodge the fact that everyone had aged: "I wanted to bring our heroes up to date. What had happened to these people, how had the relationships changed?" He gives Kirk a mid-life crisis.
Great Moments
The Kobayashi Maru test is excellent as an opening sequence, but also its use as a recurring theme throughout the film as a no-win situation. This test has inspired many ST fiction writers.
The Genesis project and the Project Genesis Tape (It took a crew of ten artists almost six months to make it!)
Spock's funeral: "there was never an attempt to create a situation to make future treks with Spock impossible, just an intent to do something different."
The Ceti Eel, ear slug parasites that are given to Chekov and Terrell.
The epic conflict between Khan and Kirk (Khan is a very good villain.)
Plot Oversights
Everyone quotes the "How do Chekov and Khan know each other?" as a major plot oversight. But although I do think the writers made a mistake here, it is not so implausible: the Enterprise is a big ship and it is not stated anywhere that Chekov joined the Enterprise in the Second Season. We know that he joined the Bridge crew in the Second Season, but he could have been working on the lower decks in Season One. Only Chekov fails to remember Harry Mudd from Season One when they meet him again in "I, Mudd" (everyone on the ship at the time would be was aware of Harry Mudd and his three beautiful women.) But Chekov could have joined after Mudd, and before Khan.
What I find more unlikely is Khan's great knowledge of the Twenty-third Century. He waxes eloquently about the far-flung places in deep space that he will chase Kirk to, and then he starts quoting Klingon Proverbs (Revenge is a dish best served cold.) This is a man who spent 200 years in cold storage, then a short time on the Enterprise, then was marooned on a desolate planet. He has never even seen a Klingon and never been anywhere other than Earth and Ceti Alpha V. The only time he had access to a library computer would be on the Enterprise.
After Spock's coffin is shot from the Enterprise toward the Genesis planet, the scene moves down to the planet. The camera meanders through the jungles until it eventually reaches Spock's coffin. In one of the jungle shots, a highway (cars and all) is visible in the lower left hand corner of the screen.
The bloodstain on Kirk's jacket left by midshipman Preston in sickbay keeps moving around and changing shape.
When he first meets Chekov, Khan grabs hold of his spacesuit and lifts him up in the air. When he lets go, he has to pull him back down again! Are there some strange anti-grav fields inside the cargo container?