Lights, camera, action!
The Search for Spock begins where The Wrath of Khan ended…but if you're a little hazy on the details, don't worry; Kirk's captain's log fills us in right away. (Kirk's a good captain like that.)
The Enterprise has finished its repairs from the battle with the USS Reliant and is heading to Earth, but the optimism Kirk felt in the light of the Genesis planet has been replaced with a sense of lose at the absence of Spock, who sacrificed himself to save the ship. (Spock's a good Vulcan like that.)
As the Enterprise enters Spacedock, someone breaks into Spock's quarters. Kirk investigates and finds Dr. McCoy acting weird, rambling on about going home to Vulcan. (Dr. McCoy's a good…no, wait. This is actually totally out of character for this dude.)
After a medic calms McCoy down, the ship docks, and Admiral Morrow comes aboard to debrief the crew. He informs them that the Enterprise is to be decommissioned. This is an all-around bummer.
Meanwhile, other happenings are happening in the universe. David (Kirk's son) and the Vulcan Saavik are investigating the Genesis planet when they discover an unknown life form. They beam down to the planet to investigate. Also, a Klingon named Kruge steals the information on Genesis, believing it would make a killer weapon—mostly because it absolutely would.
We cut back to Earth and find that Spock's father, Sarek, has visited Kirk. He demands to know why the admiral did not bring Spock's katra, his soul, to Vulcan. Kirk says Spock made no such request of him. After a cozy mind-meld session and a peek at the Enterprise's security footage, Kirk and Sarek discover that Spock entrusted McCoy with his katra. Sarek tasks Kirk with retrieving his son's body and katra.
Kirk visits Morrow and asks to borrow the keys to the Enterprise, but Morrow says "no" because Genesis has become a galactic controversy. Rather than point out how silly that reasoning is—like how there's already a ship there or that it's a request from the Vulcan ambassador—Kirk just decides to steal the Enterprise.
Sorry, sorry: he aims to commandeer the Enterprise.
After breaking McCoy out of Starfleet jail, Kirk and his crew commandeer the Enterprise, which Scotty was nice enough to reprogram for automated use. The USS Excelsior gives chase, but Scotty was also nice enough to sabotage the ship's transwarp drive.
Scotty's the man.
Checking in on Genesis, David and Saavik discover a young Vulcan boy on the planet and surmise it's Spock…or, at least, his newly minted body. It's all thanks to…science? They report this to the USS Grissom, but before Captain Esteban can contact Starfleet with the news, Kruge's Bird-of-Prey appears and blows them out of the sky.
Kruge and his away team teleport to the planet below. Genesis begins to self-destruct, and David admits to Saavik that he used protomatter in the matrix to solve certain problems (i.e. the McNugget method of cheating). Spock's rapid aging leads him into pon farr, a dangerous time for Vulcan males in their, ahem, sexual maturity. Saavik helps him through the worst of it, but the trio is soon captured by Kruge and his away team.
The Enterprise arrives at Genesis. After a game of cat-and-cloaked-mouse, it battles the Bird-of-Prey to a standstill. Kruge gains the upper hand when he threatens to kill one of the three hostages on the planet's surface. The Klingons choose Saavik, but David interferes, saving Spock and Saavik but dying in the process.
With his son dead, Kirk agrees to hand over the Enterprise but not before he sets the ship's self-destruct. When the Klingons board the ship, Kirk and his crew teleport to the planet, and the Enterprise explodes in a fiery death over the planet. You think they had it insured?
Kirk and his crew intercept the Klingons and kill them, freeing Saavik and Spock. Realizing he also destroyed their ride home, Kirk contacts Kruge and says he'll exchange Genesis for their safety. Kruge teleports down to the surface, and he and Kirk have a fistfight over the exploding planet. Kirk knocks Kruge into a lava pit and beams up to the Klingon ship.
After a daring escape from Genesis, the crew brings Spock to Vulcan where the ritual of fal-tor-pan is performed. Spock's katra is retrieved from McCoy's mind and placed in Spocks' brand-spanking new body. At first, Spock's memories are a mess, but he slowly comes to remember the crew, the Enterprise, and his friend Kirk. The film ends with the crew embracing Spock in a happy reunion.
Sure, they've got that whole treason crime hanging over their heads, but that's a story for the sequel…and nothing a couple of humpback whales can't fix.