The Christmas Episode, "On Angel's Wings". It centers around Scott and Rogue, who stay at the mansion for the holidays since they have nowhere else to go. They team up to stop Magneto and bond over having similarly complicated lives and involuntary powers. In the comics, the two are friendly coworkers at best, but the show makes a strong case for shipping them (for bonus points: this was on season 1, before Scott realized his feelings for Jean).
In particular, Scott's confession to Rogue that he used to believe in angels. One gets the impression that he couldn't tell this to anyone else for fear of looking weak, but he trusted her with this secret.
And then we have the scene where the lady that Warren saved from a fire is reunited with her daughter, who thought she had died. They hug and the lady tells her kid that an angel had saved her.
There's also the Kurt and Amanda subplot in "Shadow Dance", where Kurt tries to hide his mutation from the girl who asks him out... but it turns out she knows he's a "blue fuzzy elf" and is okay with it.
Kurt: You won't like what you see.
Amanda: Let me be the judge of that.
Kurt: (turns his image inducer off and stands there hanging his head)
Also from "Shadow Dance": Todd and Freddy bemoaning how no one has asked them out, then Tabitha comes in and tells them she's taking both of them as her dates.
The scene at the end of "Power Surge" when Jean says to Scott after he helped her come out of the Unstoppable Rage her evolving powers forced her into, "That's what I like about you, Scott. You know me better than anyone ever has or ever will." Ship Tease anyone?
The plot on Mystique's side in "Shadowed Past". After a visit from Professor X, she decides to tell Kurt that she's his mother. The only reason behind this was because she was reminded of losing her son when he was an infant and wanting him to know his origins. She was even upset when he recoiled in horror and angry that Magneto had sent his team to interfere.
"Blind Alley", where Mystique kidnaps Scott and drops him in Mexico without his glasses. She attacks him in a warehouse only for Jean to get a Moment of Awesome and save him.
Which itself is a CMOH for Logan, since this is literally the only version of the X-Men where he'd say that instead of a jealous "he doesn't deserve her" rant.note In the comics, Logan has a crush on Jean because she resembles his first girlfriend, Rose. He does resent that she takes Cyclops over him, but takes it the saner way. Besides, in this show, the age gap between them is already enough that it would make a potential relationship between them creepy.
It's also kinda funny since it's almost exactly the same thing he said in the comics when he realised that Piotr and Kitty had finally done the deed.
"Grim Reminder", where Wolverine has had a chip implanted in his head that drives him insane. Kitty goes up to him anyway and is able to get him to remember her. At the end of the episode, we find out that Kitty has been checking on him in the hospital every hour to see if he's alright.
Logan and Rogue together full stop. In "Self-Possessed", where all the people she's ever absorbed overpower her and Logan is the one who convinces her to fight them. At the end, she comes out of her unconscious state to hold Logan's hand and smile up at him.
In "Turn of the Rogue", Wolverine asks where her allegiance lies. Terrified and shaking from the cold, Rogue asks if she'll be thrown out of the jet if she says no. Not only does Logan tell her no, but offers her his hand after she says she'll join. D'aw.
Lance saving Kitty from the falling stands.
Any time Kurt & Kitty interact is either this or a funny moment. In the episode "Rogue Recruit", previously Kurt had tried to impress Kitty and just freaked her out. When she wants to prove herself to the others after Logan tells Kurt to watch the "half-pint", she ends up freaking Rogue out even more and getting Nightcrawler hurt. After pleading for him to wake up, ("Come on Kurt, wake up! Tease me! Scare me! Anything, just be OK!"), she manages to contact Xavier and get Kurt help. When he recovers she tackles him and crushes him with hugs while calling him a fuzzy elf. This caused lots of fan fiction on the internet.
Kurt being at his sister Rogue's bedside, even just to be there to comfort her after the events of her power surge. He then happily tells her, (because Kurt found out a few seasons ago that he's Mystique's son and now Rogue found out recently, she's Mystique's daughter-through adoption) that he's so excited, because Rogue is basically his big sister now, and that he always wanted to have a brother or sister. AWWWWWW!!!!
Rogue's dopey smile after Kurt teleports out of the room, showing she's touched by his concern and enthusiasm at being her brother.
In "Impact", the rest of the brotherhood had been screwing around with Mystique, now a statue, and were even going to convert her into a fountain. Wanda was already a tad disgusted by this, and when Kurt shows up, she prepares to attack him until she learns the real reason for his presence.
Wanda: "You're in the wrong house, X-Man!"
Kurt: "Wait! She's- She's my mother..."
Wanda's expression immediately turns to shock and then remorse, and she leaves the room without a word, letting him escape.
She later agrees to arrange a meeting with Agatha Harkness, one of the few people who might be able to restore her.
Kitty rescuing Danielle Moonstar from the underground cavern in "Ghost of a Chance". Very few people believed Dani even existed, and no one believed she was still alive. Everyone figured that Kitty was going insane when she continues to talk about her dreams about Dani. And yet she continues to search for Dani and eventually saves her from a terrifying death via drowning.
From "Operation Rebirth", the episode in which Captain America guest stars, the bit where Wolverine tells a cryogenically frozen Captain America how they made a good team and how Wolverine will find a cure for his condition.
Also in that episode, Logan saves several people from a Nazi camp. Including a teenage boy, who then tells him this: "My name is Erik Lensherr. Thank you for saving us."
And Magneto subsequently releasing Wolverine with the line "A little boy in Poland owes you that much."
Whatever happened before or after, the scene where Pietro talks to Wanda in "Hex Factor" is heartwarming; he starts by complimenting her haircut — which, let's face it, not a lot of brothers would do — and tries to apologize for her time in the asylum. It rang true for this troper, who has siblings, and probably for many more.
"X-23" has Logan learning of the utter hell, well, X23 was put through, and becoming utterly enraged over it. When she comes after him he finally manages to fend her off long enough to explain he's also a victim, as she breaks down crying over everything that was done to her, before Shield arrives and despite her wanting to stay with Logan he tells her to run, before he stands up to Nick Fury and tells him to let her go.
She returns the favor in "Target X", when Hydra hires two bounty hunters to go after Logan, knowing she will find him. With them on their tail, X23 repeatedly tries to get rid of Logan, not because he's slowing her down as she claims, but because she intends to be captured so she can blow up Hydra, and knows if Logan follows her he'll end up dead.
Toad, yes Toad gets one in the episode "The Toad, the Witch and the Wardrobe": After rescuing Wanda from Mastermind and her own father she gives him a kiss (though he was in the form of a ski instructor thanks to Kurt's holographic inducer at the time, but still). After being told that he put his life on the line for her, she gives him a soft, grateful look followed by a very genuine "Thank you". Then later, at the Brotherhood of Bayville Boarding House, she actually talks to him in a casual manner about not being mad at her father anymore.
Spyke's documentary about the X-Men, especially when he calls them his new family.
After being an almost irredeemable Jerk Ass for much of the series, Pietro leads the Brotherhood into battle in order to help his sister and the X-Men save his father from Apocalypse.
The end of "Uprising", as Xavier tells Leech's mother that Leech has a place where he will be welcomed... when she feels he is ready to go. It's nice to see Xavier offering a hand of support to a mother who had been paranoid about the fact her son was a mutant.
That, and the X-Men risk the obvious backlash as there's protesters at the mansion itself to help out one of their own.
Another one from "Uprising" that has surprisingly gone unmentioned until now. Leech's powers go overload and suddenly everything in sight loses power. The lights, the mutants, everything. Then, the light behind Scott's visor goes out. Jean removes it, and, for probably the first time ever in this version, they can fully look into each others eyes. *sniff*
"The Cauldron" for Scott and his brother, Alex. After ten years, the two discover that the other was alive and they immediately rush to each other, even Alex giving Scott a brotherly punch (payback for all the hard times Scott game him back then). However, Alex was reached by Magneto first, and Scott (unaware of the details on Magneto) is hesitant, but follows Alex through and sticks with his brother. Throughout the episode, you get the vibe they are a family and they've finally found each other. And at the end of the episode, the two are in free fall again as Asteroid M explodes around them. They reach out for each other and proceed to use their powers to slow their landing manually. Then they proceed to completely destroy Asteroid M before it falls on everyone.
Xavier gets one as well, not holding Scott's apparent treason in regards to Magneto against him and understanding that Scott was simply misinformed.
Kitty politely telling the others angry they didn't get to go to Hawaii by saying Scott is meeting his brother who thought was Dead for a decade and probably didn't want a crowd. Kurt also cheers everyone up by suggesting they go to a local beach to make up for it.
As part of the above, unlike in many other continuities, Scott and Alex proceed to stay in contact and reform their brotherly relationship even if it is over long distance as Alex is not able to go to the Xavier Institute. Alex shows up to help fight Apocalypse, and is shown in the group shot of the Xavier Institute that ends the series. "Adrift" is another great example: they are both lost at sea in the middle of a storm, but they'll be damned if they'll let each other die, no matter what the circumstances.
Scott is very dedicated in his mission to make friends. He makes a concentrated effort to befriend Rogue, even when she's being rude and clearly wants nothing to do with him (which is later rewarded with her switching sides). Although he lets his competitiveness get the better of him in "Survival of the Fittest", at the end of the episode he tells Jean that it "felt right" working with the Brotherhood and that maybe one day they'll come join the X-men.