The new Star Trek movie offers plenty moments of heartwarming, particularly the first ten minutes. * sniffStar Trek
One particular moment that always gets to this troper: The shuttle with newborn James T. Kirk and his mother is flying away from the battle, when one of the Nerada's missiles locks on to it. The missile gets within a few feet of the shuttle, when suddenly a phaser-beam from the Kelvin blows it away, letting the shuttle escape to safety. George Kirk knows he's going to die in this fight, but he is NOT going to let those bastards get his wife and son.
McCoy is about to board the Enterprise, leaving a stunned Kirk behind, when he stops, sighs and goes back to sneak him onboard. It truly demonstrates the extent of their friendship that he's willing to risk everything to make sure his friend gets to come along. (It also ends up saving them all, and Earth too.)
Spock's relationships/'moments' with his mother and Uhura.
Especially with the latter. After the destruction of Vulcan, Spock struggles to remain stoic in front of everyone, his father included, and since he doesn't want everyone to see him like that, he leaves. Uhura follows him, and they share a tender moment in the lift. In short, Uhura hugs him while Spock buries his face into her shoulder, looking like he's going to cry. Anyone who thinks their relationship is Strangled by the Red String has to consider this scene; the normally stoic Spock has no problems with breaking down in front of her, showing just how far their relationship has come off-screen.
Sarek and Spock: "Because I loved her." That is all.
The brief meeting between Kirk and Spock Prime ends with an exchange that speaks to the notion that as much as things change they stay the same...
Kirk: Where you came from... did I know my father?
Spock Prime: Yes. You often spoke of him as being your inspiration for joining Starfleet.
Spock Prime: A trick I learned from an old friend. [gives Vulcan salute] Live long and prosper.
Spock Prime quoting his death scene - "I have been, and always will be, your friend." - at Kirk.
Not just his death scene, but one of the essential truths of a life over two centuries long. Consider also that in Spock Prime's timeline, Kirk Prime has been gone for eight decades.
Also at the end, this little dialogue between Kirk and Pike when Kirk becomes the captain and takes command over the Enterprise.
Kirk: I relieve you, sir.
Pike:(smiling) I am relieved.
Spock meets Spock Prime.
Spock Prime: Since my customary farewell would appear oddly self-serving, I shall simply say... [shows Vulcan hand salute] Good luck.
Hearing Nimoy do the monologue, to the original fanfare, and then the music. Any lifelong Trekkie will tell you that that's one of the most powerful moments in the entire movie.
After the Enterprise successfully defeated Nero and the day is saved, Keenser (Scott's little alien sidekick) is shown having joined the Enterprise crew as an engineer, under Scotty no less.
Blink and you'll miss it, but after Scotty and Jim beam into the Enterprise, Keenser actually makes a small noise and looks visibly saddened.
It's small, but Kirk giving his motor-cycle-thing away to a random worker. Sure, he wouldn't need it anymore, but he didn't even ask for any money; he just saw that the guy liked it, and simply tossed the keys to him.
In the novelization of the movie, at the very end where the Enterprise is about to warp out, the transporter flares to life.... and a rather puzzled-looking beagle trots out. Most adorable punchline to a Brick JokeEVER.
It didn't actually end up in the movie, but the scene that the writers wrote for Shatner (in hopes he would do a cameo) is really quite lovely. Check it out here.
The bloopers on YouTube, when Chris Pine and Zachary Quinto perform a smooth high-five and rebound after a take is just so flawlessly bromantic that it hurts. If you look at what they're wearing and the characters involved, they just finished shooting the scene where Kirk provokes Spock into an unstoppable rage.