Stargate SG-1 has pulled off some truly heartwarming moments, but none bigger than the episode "Heroes". When Dr. Frasier dies, instead of giving a traditional eulogy, Sam Carter lists the names of the many people who are alive because of her being at the SGC.
The scene between Jacob Carter and Selmak in "The Tok'ra, Part 2". It's played almost like a romance.
Any episode with Cassandra in it. Yes, even when she's a bratty teenager.
Just about every episode Teal'c and his family are together again.
Vala's dad is a complete and total jerk, but it's sweet when he finally succeded in giving her the necklace he'd been trying to give her all episode.
It's also nice to know that she didn't throw the jewelry box he gave her as a child away after all.
Sam and Cam repeatedly making each other macaroons whenever one of them is in the hospital. They're always terrible, but it's nice to know that they're making the effort.
The episode Line in the Sand was one of the darkest points in the series. About halfway through the episode, Sam, who was shot earlier, tells Mitchell the password to a sealed file on her computer that has goodbye letters to everyone. Besides the password being 'Fishing' -a nice callback to an earlier episode- Mitchell's only response is "Aww, man, now you're gonna have to change your password!" His utter (unfounded) conviction that she would make it had me grinning for hours. She lived.
Speaking of Mitchell, they had one episode where he was accused of murder. The rest of the cast spent the entire episode trying to prove his innocence. For a lot of people, that was the moment when he was really part of the team.
"You better not be messing with me."
When Don S. Davis (General Hammond), who had been absent from the series except for guest roles for a few seasons, died in 2008, it made his previously-filmed appearance in the subsequently-released Stargate Continuum straight-to-DVD movie more poignant, especially when Carter says, "It's good to see you again, sir." Meanwhile, in the series finale of Stargate Atlantis, the Daedalus-class ship about to be commissioned for Colonel Carter's command had its name changed at the last minute — from the Phoenix (as Carter's ship in an alternate reality was named) to the General Hammond, in honor of the recently-late Hammond (and, of course, his actor).
At the end of the first episode of the second season, the team has defeated Apophis sort of and gone back to Earth. But they had to leave Daniel, injured and dying, on Apophis' ship. When they get back to SGC, though, he's already there. Sam says his name in this shocked voice, and Jack shouts "Space Monkey, yeah!" and goes over and hugs him. The final shot, of the team and just about everyone else in SGC hugging as the camera zooms out through the Stargate, had to get a smile out of everyone.
The entire team agreeing to defend Teal'c in Cor'Ai.
Believing Daniel in Politics.
Alternately, dealing with Daniel actually being crazy in Legacy.
More Danny: pretty much every sequence between him and Jack in Abyss. Later, him visiting Teal'c in The Changeling.
I don't know why, but Jack giving him back his glasses after he Descends always gives me the warm fuzzies.
Jack getting all protective of Daniel when he thought Reese had tried to hurt him in Menace.
The entire team's individual mini-monologues when Daniel was dying in Meridian.
Daniel convincing an amnesiac Vala to trust him in "Memento Mori".
The episode where the Tok'ra show up with their truth-machine and O'Neill has to admit that he cares for Carter "too much" to clear suspicions of being spies.
Admit it, you smiled when Eli gave Rush his glasses.
Chloe's interactions with her father in "Pain". She is entirely aware that he's just a hallucination. She simply doesn't care. It's especially heartbreaking when Eli and Scott literally have to drag her away from the vision to force her to get cured.
Goes Tear Jerkery in the end of season 1, with Eli and Chloe discussing their friendship.
Pathogen, when Eli's depressed mom gets on Destiny via the communication stones, gets to see his face for the first time in months and tells him how proud she is of him.
Greer and Chloe's quiet moment in Visitation was strangely heartwarming they were talking about how to execute her.
A point can be made for Rush's speech to Eli in "Resurgence" being this - it's rare to see Jerk Ass Rush respecting someone.
In "Hope", Greer almost immediately volunteers to donate his kidney to save Volker, as well as his constant assurance that everything will go perfectly fine to everyone else involved definitely counts.
The real moment is just before this. Volker tells Greer about how he used to watch the starts in his backyard. What does Greer do? Takes Volker to the domed hydroponics lab, the best place on the ship to watch the stars. That, more than any other moment, shows that Greer really cares for the crew.
In Epilogue seeing the development of Novus from a small settlement to a prosperous metropolis fit the bill. For all of their differences, their squabbles and the adversity that they faced, the crew of Destiny were able to band together and create a future for themselves and their descendants.
Also, Eli Wallace apparently does not care that a problem has stymied Earth's brightest minds for decades. He's going to cure Lou Gehrig's Disease. Why? Because T.J. has it.
The alternate timeline where they tried to dial in a star. Rush makes a short speech about the importance of Destiny's mission, and asks for volunteers. Young has already agreed to stay with him, and nearly half the crew volunteer as well. It actually brings Rush to tears. Subverted moments later when Telford starts yelling how they're just going to get themselves killed.
Telford: I have half a mind to force you all back.
Greer: I'd like to see you try.
Young's speech about family in the final episode, "Gauntlet", as the final group of the crew - our main characters - prepare to go into the stasis pods for three years, or possibly forever. He refers to Scott as a son, Chloe as a daughter, Wray as a sister, TJ as the most compassionate, Greer as the bravest, Eli as the smartest... and Rush as the crazy Uncle "who, despite everything, manages to come through for you in the end."
The climax of "Light". Having spent an episode resigned to their fate, the elation of the crew as they realize that they're going to live is truly amazing. For the first time since they arrived, everyone is smiling.
"Sabotage". The simple act of returning Rush's lost glasses.
The appropriately named episode "Hope" is just covered in these, the most prominent being Greer saying he'll give up a kidney to save Volker, who's dieing of renal failure without even giving it a second thought.