Enterprise, crippled after its encounter with Khan, limps back home to Space Dock in orbit above earth. There will, of course, have to be an official inquiry while they decide whether it's worth it to send Enterprise to the body shop or just pay out blue book value. Thus, the crew is given unpaid leave (or something) and told to wait. Scotty, however, is reassigned to Excelsior, where he is to help out with something called "transwarp" drive.
Meanwhile, Kirk goes back home and gets a visit from Spock's father, Ambassador Sarek, played terrifically (as always) by Mark Lenard. It seems that Spock is only mostly dead: he left his katra (spirit/soul) with McCoy who is having an allergic reaction thereto and trying to charter starships to take him back to the Genesis planet (which incidentally is now off limits).
So Kirk et al decide to steal Enterprise and take it back to the Genesis planet and try to reunite Spock's katra with his body. There is, of course, the small problem that they're in space dock and even if they make it to the Genesis planet, the science vessel Grissom is on station and it is revealed that the technology is unstable, which will eventually cause the planet to explode. Nevertheless, in some of the best scenes of the movie, Kirk et al. succeed and make it to Genesis, only to discover a Klingon bird of prey, captained by Christopher Lloyd, is already there and causing problems. In short order, Grissom is destroyed, Kirk's son is killed, an adolescent Spock has sex with Saavik, the Enterprise self-destructs, they find Spock's body (now all grown up), and kick the Klingon commander in the face. They also hijack the Klingon bird of prey and make it back to Vulcan where Spock's katra is re-united with his body and our gang requests political asylum. Yay.
The Search for Spock began the long Star Trek tradition of strange casting and/or directorial decisions (and also suggested that the odd-numbered movies being less good than the even-numbered ones might be a pattern). The replacement Saavik seemed to have been told that she must speak in a monotone and E-NUN-SEE-ATE every word, thereby turning every moment of what had been a perfectly good character into something cringeworthy. Speaking of cringeworthy, the captain of Grissom was also kind of strange and unctuous and smarmy, more like a game show host than a starship captain. (Captain Styles was over the top, but then he was supposed to be). And Christopher Lloyd as the Klingon commander was just distracting. I kept expecting him to mutter something about 1.21 gigawatts...
On the other hand, Scotty had some really good lines, as did Uhura.