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.
Also from the chapter, Hagrid's gift to Harry:
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.
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: I’ll use the invisibility cloak, It’s just lucky I got it back. Ron: But will it cover all three of us? Harry: All—all three of us? Ron: Oh, come off it, you don’t think we’d let you go alone? Hermione: Of course not. How do you think you’d get to the Stone without us? I’d 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.
And later in the same chapter, Ron sacrifices himself in a game of chess so that Harry can go on to stop Voldemort. Keep in mind that these are eleven-year-olds.
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.
In The Philosopher's Stone, 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.
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."
The Mirror of Erised. Enough said.
"Mum?" he whispered. "Dad?"
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.
From the Film:
The book and film of Philosopher's Stone both have this scene, but the acting by both the late great Richard Harris and young Matthew Lewis puts the filmed version ahead in my opinion.
"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 example from Philosopher's Stone?
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.
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.
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.