The relationship of Po and Mr. Ping. They have a genuine loving relationship. The fact that Po is a panda and Mr. Ping is a goose is not even a reason to question that they are father and son. It's a joke in the movie but it's still touching.
When Tai Lung invades the village and Po believes he is a failure, Mr. Ping tries his best to comfort Po, telling him he still has a chance with his noodle business.
Oogway is completely confident that Po landing in front of him during the ceremony was no accident and tries to assure Shifu of that till his end.
Shifu leads Po to the Sacred Pool of Tears. Remember, until that morning, Shifu largely treated Po as an intruder who had no place in his world. Now at that moment, Po realizes that the Master has just led him to the holiest place of all of kung fu tradition with the red panda asking "Do you want to learn kung fu?" When Po answers "Yeah! ...OKAY!" with all his heart, you can't help but tear up with Po when Shifu responds, "Then I AM your... master!" to show that he knows that the Panda is not a worthless failure, but someone whom that teacher has faith in at long last. But what Shifu says immediately after this is kinda funny.
Shifu: (deadpan) Don't cry.
Po: Okay. (sniff)
And that all pays off when Shifu and Po fight over the dumpling, midway through, Shifu pauses with a look of wonder at Po moving like he was trained to fight all his life and realizes that the giant panda is the Dragon Warrior after all.
More important is that Shifu is smiling, with brief bits of laughter, while fighting over the dumpling. He's rediscovered the joy in teaching martial arts.
"I'm not hungry." Followed by Shifu and Po bowing to each other. You can see it in Shifu's face that he finally believes Po can be The Dragon Warrior.
The dinner scene where the Furious Five, although still understandably resentful of being upstaged by Po at the choosing ceremony but have grown to respect his tenacity, all finally begin to befriend the affable panda. They also all start cracking up (except for Tigress) at his impersonation of Shifu.
Except for Tigress, who stoically eats her tiny plate of tofu. She doesn't really accept him until the very end of the movie, in which she is the first to respectfully bow and declare him a master of kung fu. She doesn't seem to have as much anger and resentment towards him as she did before though so she was at least able to tolerate him at that point.
Though the very first sign could be when she overheard Po doubt himself, not only was she concerned for Shifu, but possibly Po as well.
The above is probably because she realized that she was wrong about Po and his motivations. Her initial assertion that he was a devil-may-care idiot who was incapable of taking kungfu seriously. Overhearing his argument with Shifu, where he reveals his crippling self-loathing, proved to her that this actually means everything to him, and that he saw being Shifu's student as his only real shot at being worthwhile in any capacity. Knowing her backstory, it turns out they're Not So Different.
Viper, on the other hand, hides any resentment on her part very well, being the only member of the Five to treat Po with unconditional kindness from the moment she met him, even showing concern for his well being when he's not on camera. She's the nicest to Po because, as the only one in her family born without venom or fangs, she knows exactly what being an outsider feels like.
Mantis begins to treat Po with respect for similar reasons.
Mantis: Who am I to judge a warrior by his size. I mean, look at me...
After Po gets tossed down the stairs by Shifu, Tigress commented that "If [Po's] smart, he won't come back up those steps." Monkey responds by saying Po will, making him the first to acknowledge Po's unbeatable spirit. He's also the first member of the Five to actually call Po by his name.
When Shifu decides to stay behind in the valley to take on Tai Lung, Po seems like he won't leave. Crane kindly puts a wing around the panda and leads him out, understanding his loyalty to Shifu. And when you remember the complete awkwardness during their first interaction, the compassion Crane demonstrates here shows that he has have come a long way in understanding Po.
When the Five are evacuating the villagers, a little piglet is walking around Master Monkey's legs, obviously lost and confused. Monkey gently picks it up, holds it close, and says, "Come little one, let's find your mama." It's small, but it makes the warm and fuzzy feelings come in droves. Made even more so by watching Secrets Of The Furious Five, in which the five overcome various obstacles and Monkey's flaw is revealed to be lack of compassion.
The scenes of kitten Tai Lung when he was first being raised and trained by Shifu were absolutely, undeniably adorable, enough to make a heart of stone melt. (Which only made Shifu's flashback to him leaping in the air for a hug, just before adult Tai Lung, lost to rage and insanity, was about to smash him to a pulp, all the more of a Mood WhiplashTear Jerker.)
The end credits are a series of these: Po showing the Dragon Scroll to the children of the Valley, Tigress doing her own impersonation of Shifu, which pans over to show Shifu laughing, Po getting an action figure of himself, and the last scene where Shifu and Po share their dumplings next to the sprouting peach tree that Oogway planted before leaving. It's all note perfect.
Shifu may be a master, but Ping wipes the mat with him at Chinese Chess.
Po adding himself to his Furious Five action figures.
I'd throw in the music that plays with it, both the traditional Chinese music and the remake of "Kung Fu Fighting".
When Shifu declares that it's time for Po to take upon the Dragon Scroll.
Po: You really believe I'm ready?
Shifu: You are, Po.
Keeping in mind that this is the first time Shifu calls Po by name, instead of just "Panda."
This is a movie with An Aesop about being yourself that actually pulls it off. The whole damn movie's basically one big CMOH.
"To make something special, you just have to believe it's special."
"There is no secret ingredient. It's just you."
The Dragon Scroll is a CMOH as stated numerous times on this page. It's supposed to contain what makes you the Dragon Warrior, the greatest fighter to ever exist in a single scroll. When it's opened... It's blank. Reflective. Po realizes what this means later on after he learns that the Secret Ingredient soup doesn't really have any secret ingredient. It's you. YOU are the only person who can make yourself special. There's no special training, there's nothing you can do but be yourself. Only you can make yourself awesome and the sooner you realize that the better.
When Tigress goes off to fight Tai Lung herself, and is followed by the other Furious Five. It just shows how close they are with each other.
Tigress: Don't try to stop me!
Viper: We're not trying to stop you, we're coming with you!
Monkey:*Thumbs up, smiling*
The scene with Po and Oogway at the peach tree is pretty touching. It starts with Po being mortified that he's eating sacred peaches and quickly spitting them out, apologizing for it. Oogway tells him that it's alright, then asks Po why he's upset. Realizing that their was no use lying to Oogway, the panda spills his heart out to the old master, revealing how everybody hates him, that not even he believes he's the Dragon Warrior and wondering if he should run back to his old life. After this, Oogway gives him some inspirational advice about not letting the pain of the past or fear of the future affect him, then taps the tree, knocking a peach into Po's hand. It's a small and simple act of compassion, but it's enough to convince Po to stick around and endure the abuse Shifu gives him the next day.
Even before he fully embraced his son's destiny as the Dragon Warrior, Mr. Ping was never harsh or mean in his assertions of Po taking over the noodle shop. He genuinely believes that Po can find the emotional satisfaction he's looking for in being a restaurateur, and is genuinely sad for Po when it seems like his kungfu aspirations have gone up in smoke. Even when he initially failed to rocket his way in to see the Dragon Warrior ceremony, Mr. Ping just sadly holds up the apron, clearly feeling sorry that Po didn't even get to see the Dragon Warrior.
Master Shifu smiles and laughs during the dumpling fight. One has the feeling he hasn't enjoyed himself in a long time, likely since Tai Lung turned evil.
The last time was during Tai Lung's training. You'll notice he's much more dour with Tigress during her training, and his nitpicky criticism of the Five during their ambush training.
A couple blink and you'll miss it to show that Tigress isn't always the tough stoic warrior that she seems to be. One is that she & the rest of the Furious was going to reach out and help Po get out of the obstacle course before Shifu stopped them. And another when Po's finishing move to defeat Tai Lung caused a huge gust of wind, we see Tigress holding a baby rabbit and she's holding it close to protect it.
The fact that all animals, no matter how big or small, can be Kung Fu masters is heartwarming not just for this movie, but the entire franchise.
The fact that everyone is afraid of Tigress because she has no idea of her own strength (being a tigress cub among ducks and rabbits), and they won't even play with her. When Shifu comes, he brings a game with him, and later adopts her because no one else was willing to. Considering the last big cat Shifu trained...
Viper and her father smiling at each other (it's significant, alright) after Viper had defeated the gorilla rogue.
Monkey's backstory, particularly after Oogway saved him from certain death. That Monkey is the only one who uses a staff also becomes heartwarming, considering Oogway uses one too.
Oogway not only shows the obvious trait of compassion in that short, he shows great humility by straightening out a small-town prankster—a problem which most would call too insignificant for a grandmaster's attention.
Mei Ling in Crane's backstory gloriously averts the Cats Are Mean trope. In spite of the fact that she's the best student at the academy (and Crane is only a janitor), she's the first to see Crane's potential. She's the only one nice with him at the kung fu school and she encourages him to train and unlock his inner potential.
Master Shifu comes to the restaurant and finds that true perfection he wanted at the banquet is at the informal dinner at Ping's restaurant; where the lonely have a place in a holiday where togetherness is everything. When Shifu turns to leave, feeling he does not belong, Po comes out and tries to persuade him to come in.
Shifu: Oh no, I couldn't. These are your people; this is your family. Po: You're my family too.
Tigress has SEVERAL. Wishing Po a happy holiday with a genuine smile was one, feeding a baby piglet playfully another by saying, "I'm gonna get you!". Considering how much of an anti-social jerk she was in the first movie, these examples only turn the D'awwwww factor up to twelve and begin the Po/Tigress Ship Tease.
That and Tigress reminiscing with the others at the formal party how much she enjoyed the holiday when she was a child. Though she wasn't the only one, the other masters joined in pretty quick.