Stitch and Angel's relationship. Angel's feelings for Stitch cause her to reform and when they're separated it deeply hurts both of them. When they're reunited, Stitch confesses his love for her, which breaks her out of her Heroic B.S.O.D..
The entire concept is kinda touching: Lilo, who was able to turn a hostile monster who was created to destroy, into her best friend within a few days. Now her job is to reform all 625 of Jumba's other experiments. While it's hit or miss with some, she does genuinely care about the experiments and sees them as Stitch's family. She's essentially trying to help her best friend's family get into a better life.
Lilo's support of Stitch when he gets framed for causing destruction around the house, even after finding her record player broken, shows just how much she trusts in him.
Mr. Stenchy being adopted by a couple on Pleakley's home planet, since his smell reminds them of their first date.
What is Stitch's first idea with Cannonball's powers? To help Nani from losing her job. It seems that he doesn't just love the youngest of the two Pelekai sisters.
Despite the fact that Mertle is... Mertle, it is shown that she does genuinely love her dog/experimental alien Gigi and is sadden when she thinks she is gone.
Stitch holding himself back from causing destruction so he can help Lilo.
In "Amnesio", Lilo and Gantu, having suffered amnesia, surprisingly make a cute team-up. Lilo even makes him a pot that says "World's Greatest Cop". Aww...
The end of "Fibber", where we see that Pleakley's family truly does love him despite his faults.
Lilo goes all out with wanting to help her "aunt" impress his family, since she does care about him.
Lilo and Stitch apologizing after their short problem with each other, and even Lilo's friends acting a tad nicer for once. Shows that if Mertle didn't do a lot of the controlling, they could potentially be real friends.
Elastico's Circus Family. They are very accepting lot of people who took in the alien monster with the Rubber Man powers. He was able to find his one true place on his own, just like Stitch.
Instead of outright insulting Lilo for being an odd kid, the politician that visits her town actually listens to her (while not believing her) and humors her with having Yaarp be the city's official alarm for alien invasions. A guy like him could have easily just not listened to her, but still did it.
When the alien that lives on said asteroid causes a rock slide to fall upon our duo, the first thing Stitch does is fall on top of Lilo to protect her.
Jumba accepting that Earth is his home now after deciding to sacrifice the hyperdrive to push the asteroid away from the Earth.
Keoni giving Lilo the "Mud Monster Hula girl" toy that he likely made himself to cheer her up since it was clear she seemed upset. Shows that he does care about her, even if it's just as a friend.
When Stich presents Hämsterviel to the Grand Councilwoman, he is sure to let it be known that it was Finder who found the fugitive. The Grand Councilwoman, though, points out that, even though Finder tracked down Hämsterviel, Stitch was the one who brought him in, and then notes that it is good of Stitch to share the credit regardless.
"Dupe" has a genuine moment of friendship between Gantu and 625; Gantu apologizing for his behavior and taking 625's feelings into account and 625 listening to Gantu's Troubled Childhood and telling Gantu he's not offended by being called a "trog". As someone who calls 626 and other criminals this, it's very sobering.
Lilo finds out that having Stitch being her friend is enough for her. True Companions at its finest.
Lilo reassuring Stitch about his height, and the guy who didn't let him on the other ride allowing him to go on the first day of Shortstuff's ride.
Angel's Heel–Face Turn when Stitch shows he has genuine emotions for her, as she regrets her prior actions and makes up for it by turning the cousins back to good before they could knock him out. It crosses into Tear Jerker when they are separated though.
After meeting Angel face-to-face, Stitch becomes the most gentlemanly he's ever been in the entire franchise.
He never fought Angel once and was fully-supportive of her from the start when Lilo wouldn't accept her (even though Lilo was right to do so at the time).
He excused himself for picking his nose with his tongue in front of Angel, which actually doesn't seem to offend her and she does the same nose-picking action herself, which delights him.
He wakes her up with a bouquet of flowers, and when she sings to him (not realizing what she was really trying to do to him), he compliments her. ("Nice pipes!") Although she was confused by the failure of her siren song, Angel appeared to be touched nonetheless.
In fact, it was likely this moment where Angel saw Stitch as more than just someone to use in her plans, especially in the immediate scene afterwards where she looks back at him with regret before running off to start converting the other cousins.
Lilo and Stitch are clearly concerned for Pleakley's health when Poxy infects him, going very far to keep him and the experiment safe from Gantu until they can get it out.
625/Reuben takes care of Gantu when he ends up getting sick from Poxy instead. Shows that he does care about the big lug and Gantu is somewhat appreciative.
While Lilo's experiment-placing abilities are largely hit or miss, she couldn't have made better use of Babyfier's ability to turn people into babies. The episode ends with him set up in the animal shelter, turning old dogs slated for euthanasia into puppies. In other words, she's reviving old dogs that can go back to being cute puppies.
Baby!Nani hugging Lilo. You gotta admit, this helps Lilo really appreciate her big sister seeing her like this.
"Bonnie & Clyde"
Despite being criminals, the titular experiments share a close bond like their namesakes and it seems Clyde is concerned for Bonnie when it comes to them finding Lilo and Stitch and getting them involved with their crimes.
Pleakley taming Spike with a hug, which touches the other experiments in his therapy group. Even Nosy cries.
While his powers make people incredibly silly when they get hit with his quills, Spike is really an affectionate creature after E.A.R.W.A.X. He even wanted to give a depressed Lilo a hug to cheer her up.
Nani going full Mama Bear in this episode. Seriously, she's dealing with a man-eating alien who stuffed her sister and the semi-indestructible monster to the point that they cannot defend themselves, but she goes against it armed with only a broom. Shows how much she cares about her 'ohana and is willing to defend them when they cannot do so themselves.
Victoria accepting Lilo for who she is. While she isn't around much, it's nice that lilo makes a new friend.
Despite his powers, Lilo and Stitch care deeply about the little guy and personally wishes to help him despite his the fact that he's a walking bad luck charm.
In "Checkers", Nani is overworked in the beginning. When the Pineapple Princess calls in sick for the float, who does she decide to call in to replace her? Her older sister, who has always been there for her.
Nani reunites with some old friends who have succeeded in life, despite one needing crutches when they first see each other again. Despite that, her friends do think that Nani is still doing good since she's a homemaker and likely doing very well for her situation. They don't look down at her for having a normal job.
Trixie acting as a Cool Big Sis to Lilo and giving her life advice. She may be the Sassy Black Girl in her own show, but she's still a nice person and helps the young girl even when she puts into something. She even calls out on Spud and Jake for encouraging her bad idea.
While a little misguided, Lilo wanting to get Keoni the skateboard is kinda sweet in a way.
Lilo reassuring Penny about her article. Lilo is a sweet kid and Penny appreciates it: considering that the latter, in her own series, has to deal with unsupportive friends on a regular basis, you can only imagine how much it means, to her, having someone who genuinely cares. Sure they get under Spats' spell, but for the most part, they are getting along great.
Oscar and Trudy do have a nice moment or two before Spats gets involved. While they're a kinda harsh couple, they do care about each other.
During the comedy routine, Pleakley is ecstatic when he finally gets his comedy act together and everyone starts to have fun. While it's at Heckler's expense, they all seem to enjoy poking fun at themselves to help how fun the community is.
Kim being concerned about Lilo's safety, but gives her an impressed tone when she sees how well Lilo can handle herself. It shows that an Action Girl like her still has her inner baby sitter being a caring person around kids, so it makes sense why she's so concerned about her safety and keeps her away from them.
While it's silly, the 3rd Street Kids under Lax's spell are just having fun with Stitch. It's kinda cute to see them all having fun despite that. Even a little Ship Tease with T.J. and Spinelli.
Lilo meeting an entire group of her (more or less) peers as unconventional as she is. Sure, it's not explicitly stated anywhere in the episode, but just look at how fast the 3rd Street Kids end to getting along with her, no mention to the fact they immediately believe to all the aliens and experiments' matter (something neither Kim nor Penny did; Jake doesn't count due to his mystic nature) and that are totally willing to help her with it. Really, that must have been one of the happiest moments in Lilo's life: discovering that there's more than one person who sees things like she does.
Crossing over into Tear Jerker is "Remmy", which takes place on the anniversary of Lilo and Nani's parents' deaths. Lilo is having nightmares, courtesy of the latest experiment, and Nani has Pleakley enter her dream world to give Lilo the photograph of them and their parents. Lilo remembers the day it was taken, and the experiment loses its power. After Jumba, Pleakley, and Stitch successfully capture it, they turn to see that Lilo is with her family, whole once again.
"Mrs. Hasagawa's Cats"
The fact that Mrs. Hasagawa, despite being a Scatterbrained Senior, is a very caring woman who took in these creatures and has gotten their habits down to a tee. It shows how caring she is and how good of a caregiver she is.