Star Trek VI gave Kirk a few sweet moments: who can forget his triumphant "Fire!"?
Also, the very end of the film, Captain Sulu bringing in the cavalry. "Fly her apart, then!", not to mention "Target that explosion and fire." when he arrives. It's the closest you ever get to Sulu saying "That guy, ruin his shit." The man has a half-dozen lines and barely a minute of screen-time in the film and he still manages to be an equal badass to Kirk. As stated elsewhere on this wiki, many of Kirk's crew closed out their careers ranked as captains, but only Sulu became a Captain.
Sulu gets his awesomeness in from his very first scene, where Captain Sulu calmly sips his tea on the bridge of his ship, with the smuggest grin imaginable on his face. He's been pining for Excelsior for the last three movies, and now he's got his wish.
And he was even supposed to be the straightforward Big Damn Heroes at the end, until Shatner pitched a fit about the Enterprise having to be saved by another ship.
Also, steering Excelsior into the Praxis shockwave and allowing the ship to ride it out. Awesome.
Basically all of the main cast get something to do in the final climactic scene at the Khitomer conference.
Kirk, of course, tackling the Federation President out of the line of fire.
Uhura, McCoy and Chekov clearing Kirk's way to do the aforementioned Diving Save of the President.
Chekov stops the Klingon "Ass" in Ambassador (from this film and The Voyage Home) in his tracks. Not that the ambassador was involved in the conspiracy, but still. In the background, the Starfleet Commander-in-Chief, "Bill", can be seen calmly strolling towards the chaos to help out.
McCoy arrests the Romulan ambassador, Nanclus, who conspired with Chang, Cartwright and Valeris (with Sarek looking on from behind).
Spock confronting Admiral Cartwright with Valeris and her incontrovertible proof of his own involvement, prompting a visible Oh, Crap! from Cartwright.
Scotty kicking a metal door down for a Dynamic Entry and phasering the "Klingon" sniper (who turns out to be Colonel West in disguise) out the window to his death.
Cartwright attempts to flee, whereupon Sulu and two of his own crew from the Excelsior beam down and arrest him as well. Sulu's line is also dripping with venom: "Cartright, just a minute."
Points also go to Chancellor Azetbur's guards, who immediately surround her and put their lives on the line for her safety as soon as the sniper shoots at the President.
The Klingon ambassador applauding Kirk's speech, after having called for his head both earlier in this movie and in Star Trek IV.
Noteworthy as well, because he BOLTS to his feet the moment that torpedo launches - he instantly recognizes that the Enterprise figured out where they were, and have won, despite the killing blow not yet landing.
Speaking of that, Spock and Dr. McCoy modifying a photon torpedo to locate Chang's Bird of Prey. They do it as it is being laid in the torpedo tube, so they had limited time—not just because of the barrage, but because of the time from loading to firing—and had to work in close quarters. With a little funny mixed in, as they kept having to listen to Chang's soliloquy. But they get it working right before it loads to be fired.
McCoy: We've got a heartbeat! Kirk: (over comms) Where's that damn torpedo?! McCoy: She's ready, Jim! Lock and load!
The entire scene where Kirk and Sulu mercilessly demolish Chang's Bird-of-Prey with torpedo barrages—very likely got wild cheers when seen in theaters too.
As scary as it is for everyone involved at the time (including him), Kirk surrendering. As much as he hates Klingons, both in the movies and in the series, it’s the final push he needs to actually be a diplomat in the quest for peace, and not the Blood Knight he was trained to be and what the people in charge of plotting still expected of him.
During the battle at Khitomer, while the Enterprise is getting the snot kicked out of it by Chang's Bird of Prey, we get a brief shot of a nameless Starfleet crewman running into a burning compartment with a fire extinquisher just before a blast door shuts, sealing him inside. The guy has about a second's worth of screentime, yet he's still the epitome of Heroic Sacrifice.
