The episode "Of Ice and Men". Despite it being a parody of The Princess Bride, it is actually a very touching moment where Stan puts aside his usual stubborn attitude and has a very meaningful bonding experience between him and Francine. The B-Story has a rather touching and awesome moment, when Toshi, at Snot and Svetlana, the mail-order Russian bride's wedding, interrupts it by admitting his love for Svetlana in fluent Russian.
The episode "American Dream Factory", when the illegal Mexican immigrants are singing "Take Me Home Country Roads", which they say is about the America they dreamed of emigrating to. Stan is moved by this and helps them escape deportation.
The flashback in "Stan's Food Restaurant". Surely I'm not the only one touched that someone would go out of their way that much just to cheer up a random depressed kid?
In the episode "Virtual In-Stanity", we learn that it's a Smith family tradition to watch old home movies of the kids on their birthdays. The first video of Steve's birthday movie session reveals that (presumably because Stan wasn't present) a preschool-age Hayley held her mother's hand as she gave birth to Steve.
Stan's speech at the end of the episode "Lincoln Lover" (after he finds out that being gay isn't a choice and that many gay people support the Republican party). Also counts as a Moment of Awesome.
The end of the episode "Rapture's Delight". At the end, after Stan sacrificed himself to save Jesus and Francine from demons (It Makes Sense in Context), we find out what Stan's personal paradise is: When he enters his personal heaven, events rewind back to before the apocalypse/it turns out to just be a dream, and Stan is living with his family, a wiser man than before.
Believe it or not, Roger has actually had a couple, perhaps most notably in "Roger Codger": at the risk of the entire family being caught and punished for harboring him, he genuinely offers to sacrifice himself to cover things up, leaving himself open for Stan to execute. Stan pulls out his gun and hesitates, obliviously noticing a tear coming out of his eye.
In the episode "Bullocks to Stan" when Stan finally man's up and kicks Bullock's ass for insulting Haley. It's actually quite touching given that it was still one of the earliest episodes.
Stan describing how much he loves Hayley while posing as his own body double:
"When you were younger, and you were asleep, he'd come into your room and put a piece of candy on your tongue so you'd always have sweet dreams. And before every mission, he kisses a picture of you..."
Though it's short lived (due to Stan geting shot in the head and forgetting about it), Stan's brief apology over treating Klaus unfairly is especially heartwarming in "Haylias":
Stan: I guess I forget sometimes that you're not just a fish in a bowl. You're a man in a fish in a bowl.
[Klaus places his fin on Stan's hand]
Klaus: Maybe I could be... a friend in a fish in a bowl.
Hayley actually taking Stan's side during the hurricane episode.
How about Stan trying to save his family throughout that episode? Sure, every attempt failed miserably and made things even worse but like Hayley said, he was really trying.
Hayley and Jeff finally getting married.
Pretty much any time Stan doesn't treat Hayley like crap (examples: Stan and Hayley bonding in "Stannie Get Your Gun," Stan telling Hayley that he should stop treating her like a child on "Stan Knows Best," and Stan vowing to care for Hayley even though there's a possibility that she may not biologically be his in "The Kidney Stays in the Picture.")
Despite all the life-threatening situations Stan is thrown into on account of Roger in "Office Spaceman", Stan still saves Roger when the alien is captured by the CIA and about to be dissected.
The end of "Great Space Roaster", where Stan of all people confirms that Roger, in spite of being a massive Jerkass throughout the series in general and the episode in particular, is still considered a part of the Smith family. Roger's genuine tears of joy that follow only make the moment more poignant.
In the episode "Widowmaker", after Stan reveals the Batman Gambit that Craig isn't dead for being a "blabbermouth", but is instead living in Laos with his new wife, due to his now-ex Julie constantly badgering him to share, Francine begins to accept that some men, like Stan, aren't so touchy-feely. As she looks away , this occurs:
Stan: My favorite color is orange.
Francine: Oh, honey, I love you too.
The flashback to Stan introducing Roger to the Smith family in A.T. The Abusive Terrestrial, he puts him in a large gift box Roger jumps out of. Francine faints in horror while Hayley runs away terrified, while an overjoyed Steve runs and hugs him.
In "Flirting with Disaster" Stan lovingly encourages a disfigured Francine to flirt with him, and while Francine tries and fails to say anything sexy Stan just steadily looks at her, smiling.
Dr. Klaustus serves as a real Pet the Dog moment for Klaus.
The ending scene in dog heaven in "Stan's Best Friend".
From "The Kidney Stays in the Picture," Stan declaring that whether or not he is Hayley's biological father (which neither he nor the audience ever finds out, though given the show's loose continuity and the fact that past episodes have shown that Hayley is indeed Stan's daughter, it's probably best to just conclude that Stan is biologically the father, since no one in-universe will remember what happened), all the love and effort he has put into raising her makes her his daughter, despite the early episodes showing that Hayley and Stan hated each other because of their political views.
Also from the same episode, Stan`s response to his daughter needing a kidney (after hearing that she passed out from drinking too much and the excessive alcohol has ruined one of her kidneys) is to drop his pants and say "I`m ready, somebody shave my groin" without a moment`s hesitation.
Steve teaching Stan to use his imagination in Toy Whorey.
In Daddy Queerest, Stan tries to get Terry's dad to accept his son after he had revealed his homosexuality to him, though it failed and Terry's dad admitted that he's not going to change his homophobic ways. Terry starts scolding Stan for causing all this trouble...until Francine tells him to stop and says that despite what Stan had done, at least he had tried to fix the mess he had created. That episode showed that she loves her husband despite his flaws. Awww.
While he goes about it in all the wrong ways, Stan's willingness to not stop until he ensures Tank accepts his son for who he is, proves that in this one instance, he actually did keep his character development from "Lincoln Lover" and "Surro-Gate".
Stan: I know how he thinks, I used to be him. Scared of the intimate man-dance.
And of course, Terry telling off his father for not being accepting of his homosexuality (also an Awesome Moment, just for the fact that it takes a lot of guts for someone who is gay to tell someone who doesn't approve of it that they're not going to change, no matter what).
The ending to "Why Can't We Be Friends?" with Steve and Snot air-guitaring.
The ending of "Can I Be Frank (With You)" is particularly noteworthy for being heartwarming and horrific at the same time. Francine, wanting to spend more time with Stan, dresses up in a man suit and infiltrates his meetings with his CIA buddy. Due to circumstances, Bullock thinks that (s)he's actually a spy, and he tells Stan to knock "him" out and drag him out to the woods and kill him. Francine tries to tell Stan that it's her, but Stan starts to think that he did something to hurt Francine. He tries to call her, and Francine's cell phone starts ringing. Stan goes completely ballistic, brutally beating "her" while screaming that Francine was the most important person in the world to him. He finally realizes that it's Francine when he tries to scalp her and cuts off the top of her costume. They work things out, but there aren't many shows that can have such a heartwarming moment that involves a guy beating the living shit out of his wife.
Steve and Akiko kissing in "Spelling Bee My Baby".
The episode "Lost in Space" can be seen as a giant one, Jeff is abducted by Roger's people and he misses Hayley, and his love for her is tested by the emperor. It turns out Emperor Zing is cheating the love test to keep his slaves, but the second test proves to Roger's people that love really exists.
To say nothing of the memories themselves.
When an alien shapeshifter offers to have sex with Jeff in the form of Hayley, Jeff is briefly tempted before he tells her that she is not Hayley and he still wants to be faithful.
The last scene of the episode has Jeff on an escape craft heading for Earth, he looks into a window and sees a vision of Hayley smiling at him.
Sinbad holding off the guards for Jeff to get away resulting in a Heroic Sacrifice. He comes back as a force ghost, happily reunites with Jeff and the two ride off into space.
In "Bully for Steve", Steve receiving his bag lunch from Francine in the most adorable way.
Francine: Stan, you might not be interesting, but you're definitely not boring.
Steve and Snot decide to make their own prom dates by using the CIA's cloning machine. They're shocked when they come out as babies. They discover they're rapidly aging and will be old enough to take on prom night. After raising them for a few days, by the prom night approaches each has come to love their date like a daughter.
The Framing Device for "Of Ice and Men". It's nice to know that someday, Klaus will finally be human again.