At the end of the series premier, the entire team stands in front of the Gate in what is an obvious, yet still heartwarming money shot.
"Cold Lazarus". The revelation that the crystal-energy being just wanted to help Jack heal from the loss of his son.
The entire team agreeing to defend Teal'c in "Cor'Ai".
The man accusing Teal'c of the murder of his father (which he admits), and preparing to execute Teal'c himself, witnesses his selfless attempt to protect the innocents from a Goa'uld attack. Upon seeing this and Teal'c's calm acceptance that he is ready accept his fate, the man spares him, telling Teal'c that the man who killed his father is dead, and that "You killed him."
The rest of the team 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".
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.
"Bane". It's not Teal'c getting Ally a new, improved Super Soaker because of their Intergenerational Friendship. It's Teal'c pulling out a giant Super Soaker of his own, turning to spray Daniel with it, and then running off to play with Ally.
The scene between Jacob Carter and Selmak in "The Tok'ra, Part 2". It's played almost like a romance.
In the Season 3 premiere "Into the Fire," Immediately after O'Neill wakes up in the cryo-chamber where the undercover Tok'ra placed him to kill the Gou'ald symbiote with which he'd been implanted, he kills Hathor by throwing her into the cryo-chamber, then rushes over to check on an unconscious Carter. When she sits up, he's so relieved he gives her a bear hug. Doubles as a CMOF when Carter asks how he is, and, still hugging her, he says, "Cold! I'm a little chilly. But I'm still me."
In "Forever in a Day", Sha're manages to send a message to Daniel through the Goa'uld hand device telling him to find her son...and not to be angry with Teal'c for killing her.
The episode where the Tok'ra show up with their truth-machine and O'Neill has to admit that he cares for Carter "a lot more than [he's] supposed to" to clear suspicions of being spies.
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. This is followed by an even bigger one from documentarian Emmett Bregman:
"Our armed forces have turned the tide of world wars. Young men and women from our great country's four corners have humbled history's worse times. We carve our thanks in stone. We stamp it into metals. We carefully tend to the vast fields where the men and women who gave their lives for our freedom now lay. More than ever in our history, we cannot fail to pass these stories of courage to the next generation. We must capture their imaginations while paying tribute to all those willing to die for the preservation of our way of life."
In Part 2, Emmett Bregman's outraged rant at the press officers trying to prevent him from filming, pointing out that he's trying to show the SGC (and all their sacrifices) in a positive light. He goes on to say that all he wants is to show the public is the truth they deserve, free from government and military lies, innuendo and omissions. After appearing somewhat self-serving in Part 1, we see that all of his press antics are because he really believes in what he does and that the press has a solemn duty to report the truth, even if they don't want them to!
The quiet moments between Jack and Sam in "Lost City".
"Avatar." Teal'c is trapped in an Unwinnable Training Simulation, and analysis from the others reveals that the program's ability to learn based on the user includes incorporating Teal'c's belief that that the Goa'uld could never be defeated. When Daniel enters the program to assist him however, they eventually meet success, with Fridge adding the heartwarming. Teal'c knew he couldn't do it alone, but doing it with his True Companions was another story.
During a season 8 season two-parter, the team (minus Daniel) get captured by a minor system lord while trying to capture the Kull warrior gunning for him. After some work they manage to convince their guard that his "god" was going to abandon him and his people. After the guard frees them, he just stands there looking a little unsure of what to do with himself now. Jack just gives him a friendly pat on the back and absently tells him "Let's go." A nice little moment.
A good moment for 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.
In "Stronghold", Landry giving Mitchell permission to use a memory recall device to show his life-long friend Ferguson some of his memories of the past year working for the SGC; memories which easily could have been his, if he hadn't taken shrapnel whilst saving Mitchell's life and developed a brain aneurysm, preventing him from getting into the X-302 program.
Landry gets another one when giving the go-ahead on the plan to rescue Teal'c from Ba'al, simply because;
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.
Vala's dad is a complete and total jerk, but it's sweet when he finally succeeded 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.
Daniel convincing an amnesiac Vala to trust him in "Memento Mori".
"Threads." Teal'c and Bra'tac are discussing who will lead the free Jaffa. While Bra'tac seems sure he will die soon, Teal'c is unwilling to admit his mentor is too old and seems sure his mentor will provide wisdom and guidance to the Jaffa for years to come.
You know that big red phone on General Hammond's desk? The President of the United States is number 2 on the speed dial. His grandkids are number 1. Priorities, people.
Just about every episode Teal'c and his family are together again.
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).