Hagrid and baby Harry. The scene is short but wonderfully fuzzy.
Cursed Child reveals that Hagrid immediately declares himself Harry's "friend" as soon as he meets a tiny baby amongst the rubble.
And the next time they meet, Hagrid brings it up, saying that last he saw Harry, he was this small, while beaming warmly under his beard.
All of Hagrid and Harry's interactions early on and during his time in Hogwarts. It shows that for the first time, Harry has a friend who deeply cares about him. Heck, Hagrid is in a righteous fury when he first physically meets Harry and discovers how he was mistreated by the Dursleys, to the point that it almost seemed like Harry was the only thing keeping the nearly 9 foot behemoth from smashing the three like insects.
Hagrid is literally seeing a small, tiny baby in blankets that has been mistreated.
McGonagall adored James and Lily as her students and was devastated by their deaths. When their child is denied his letters, she floods the Dursleys house with them and also fills the insides of eggs to make sure he gets them; the woman is determined.
Hagrid showing up on Harry's birthday. When he's not busy laying on much needed but tough exposition, he is busy making Harry feel welcome and glad and standing up for him and Dumbledore, then he gives Harry what is probably one of the best days he ever had on Diagon Alley.
Also, remember how she made Harry a Weasley Jumper. This is a boy she knows for all of a minute or two, whom she must have only heard things of through her sons... and still spent the time and effort to make him a jumper like those she makes for her family. Nevermind who he is, she is all too ready to adopt him as a fellow Weasley which does happen, in a way, after Harry marries Ginny.
A small one, when Ron accidentally admits his parents couldn't afford an owl for him. You can tell, right from the start, the Ron's lack of money is something people have always made fun of him for, and he's ashamed of it. However, Harry, as the narrator notes, sees nothing to be ashamed of, and cheers Ron up by telling him about how he had no money himself until a couple weeks ago. This is an eleven-year-old boy who's been treated like dirt by his aunt, uncle, and cousin his entire life, and when he finds someone lower on the social scale than him, he sympathizes, because he knows exactly what that's like.
Harry buying the lot of Chocolate Frogs on the train, for the explicit purpose of sharing them with Ron. As the narration noted, Harry had never had a friend to share anything with before. By ride's end, Ron's got a full stomach and Harry's off to a good start with his wizard card collection.
Scabbers biting Crabbe when Crabbe tries to steal Harry and Ron's sweets becomes better when you know that Scabbers is Peter Pettigrew. Peter seems like the kind of person who was bullied as a child, and he's consciously protecting Ron.
It gets better. Scabbers attacks after Malfoy and his lackeys were badmouthing Harry's mom. Peter took it as a personal offense.
The beginning of Harry and Ron's friendship with Hermione, when they go back to save her and she winds up lying to protect them. It's really all summed up with the classic line, "There are some things you can't share without ending up liking each other, and knocking out a twelve-foot mountain troll is one of them."
Just the fact that two eleven-year-old boys risked their lives to save a girl they didn't like. We all know how loyal to each other the three of them were in their friendship, but here we see what they were willing to do for each other even before they were friends.
In the days leading up to Christmas, Harry thinks about how he will probably have no presents, knowing his aunt and uncle. He wakes up on Christmas Day to find a heap at the end of his bed, courtesy of Mrs. Weasley, Ron, Hermione and Hagrid. He happily calls out that he got presents, even!
Harry's aunt and uncle sending him a fifty-pence piece for Christmas (worth 72 cents in the United States), Ron likes the look of it and Harry gives it to him without question.
The Weasley twins insist that everyone wears the Weasly jumpers their mum made and that Percy spend the day with them and not the other prefects because Christmas is a time for family. They spend a lot of time driving those two crazy but they do truly love and appreciate them as well.
The Mirror of Erised. Enough said.
"Mum?" he whispered. "Dad?"
When Neville comes back to the common room with a Leg-Locker Curse, Harry gives Neville a Chocolate Frog and tells him, "you're worth twelve of Malfoy." This is made more heartwarming as it was the last from a box he'd got for Christmas (he'd never had Christmas presents before), and shortly after, Neville stood up to Malfoy.
After Harry rants about what will happen if Voldemort comes back and says he's going to go after the Stone, Ron and Hermione decide to go with him without a moment's hesitation, and without even telling him first:
Harry: Ill use the invisibility cloak, Its just lucky I got it back. Ron: But will it cover all three of us? Harry: Allall three of us? Ron: Oh, come off it, you dont think wed let you go alone? Hermione: Of course not. How do you think youd get to the Stone without us? Id better go and look through my books, there might be something useful
This becomes something of a Brick Joke when Dumbledore describes how the true magic of the Cloak is its ability to cover multiple people all the way in Deathly Hallows.
Also keep in mind that said sacrifice involves letting himself be knocked unconscious when the queen beats him over the head with her stone arm. Yes, magical healing would probably fix that up quickly, but he also wanted Harry and Hermione to keep going forward. So he fully expected to be left where he was for some time.
It's better than that: this video shows that Ron could instead have sacrificed Harry and pulled off the checkmate himself... and he didn't even think of that: "It's the only way... I have to be taken."
Hermione: Harry — you're a great wizard, you know. Harry:(very embarrassed) I'm not as good as you. Hermione: Me! Books! And cleverness! There are more important things — friendship and bravery and — oh Harry — be careful!"
Keep in mind, this is Hermione talking. Her intelligence is easily her most defining trait and one of the things she values above almost anything else (to the point where she might have just as easily become a Ravenclaw rather than a Gryffindor.) Admitting that there were more important things than being smart must have been like pulling teeth, but she swallowed her pride and did it anyway because she cares that much about Harry.
Dumbledore explaining why Harry survived Voldemort's attempt to kill him:
"Your mother died to save you. If there is one thing Voldemort cannot understand, it is love. He didn't realize that love as powerful as your mother's for you leaves its own mark. Not a scar, no visible sign... to have been loved so deeply, even though the person who loved us is gone, will give us some protection for ever. It is in your very skin. Quirrell, full of hatred, greed, and ambition, sharing his soul with Voldemort, could not touch you for this reason. It was agony to touch a person marked by something so good." Dumbledore now became very interested in a bird out on the window-sill, which gave Harry time to dry his eyes on the sheet.
Bonus points - Dumbledore being Dumbledore, he most likely became "interested" in that bird on purpose...
Hagrid's gift to Harry at the end of the book:
It was full of wizard photographs. Smiling and waving at him from every page were his mother and father. "Sent owls off ter all yer parents' old school friends, askin' fer photos... Knew yeh didn' have any... D'yeh like it?" Harry couldn't speak, but Hagrid understood.
From the Film:
The unmistakable look of pride on McGonagall's face as she introduces Harry to Oliver Wood as Gryffindor's new seeker certainly qualifies.
A couple of deleted scenes had their moments too such as after the Troll scene.
Harry: (To Hermione) It's good of you to get us out of trouble like that.
Ron: Mind you, we did save her life.
Harry: Mind you, she might not needed saving, if you hadn't insulted her.
Ron: What are friends for?
Harry's reaction to receiving presents on Christmas at Hogwarts. Considering how the Dursleys most likely never gave him a decent gift all those years, this was definitely a big deal.
The Mirror of Erised scene, even more so than in the book. After Harry sees his parents, what does he do? He runs to share the moment with his best friend.
The book and film both have this scene, but the film version is made better by the acting by both the late great Richard Harris and young Matthew Lewis.
"There are all kinds of courage," said Dumbledore, smiling. "It takes a great deal of bravery to stand up to our enemies... but a great deal more to stand up to our friends. I therefore award ten points... to Mr. Neville Longbottom!"
This becomes much more poignant after the seventh book reveals that Dumbledore himself didn't have the courage to stand up to his friend when he needed to the most. So when he sees that courage in a first year, he's making sure to encourage it.
The fact that, even after the Trio's heroic acts, Dumbledore arranges his speech so that it's Neville's comparatively small ten points that swing the house cup in Gryffindor's favor. Given that Newt seems set to eventually play a crucial role in stopping Grindelwald but doesn't get the credit that Dumbledore eventually does, you have to wonder in hindsight if Dumbledore made it a point to see that an unsung hero didn't get lost in the shuffle.