Charles and Raven's first meeting as kids. "You never have to steal again." Then that smile of hers! The filmmakers seem to go out of their way to cast the cutest child actors possible to maximize the "Awwww!" factor for this scene. Cue a decade and more of foster-sibling bonding.
Charles reading to Raven a page about evolution, as listening to science puts her right to sleep.
When the Man in Black first learns Charles and Raven are mutants, he's never shown to be afraid or disgusted. Then he's in awe of them whenever they show off their abilities.
When he learns Hank is a mutant, he's only upset that Hank didn't tell him himself.
Charles and Erik's first meeting. Erik magnetically-attached himself to Shaw's sub, prepared to die in his attempt on Shaw's life, when Charles—not knowing Erik at all—dives in to pull him back. Charles reveals to Erik that he is a mutant, like Erik, which leads to this exchange:
Raven and Hank's first meeting. It's obvious that he's nervous about showing off his ability, but just look at the Adorkable smile on his face when he realizes that Raven isn't turning away in disgust.
Both Charles and Erik's reactions whenever they encounter a new mutant and their abilities. Charles usually smiles with delight and Erik just looks at them in awe. You can easily see why they became the leaders and mentors of mutants!
Another little moment when the teenagers were just sitting around talking about their powers, it marked the first time this group of young people got to be surrounded by other people just like them. They spend the next few minutes hanging out and showing off the same powers that made them feel alone and like freaks. Next scene, they're partying. Beast is hanging from the rafters, Darwin is playing 'Hit me' with chairs, Angel is flying around and Mystique is cheerfully drunk.
During the showing-off scene, when the group is still deciding codenames, Havok is teasing the woefully (but understandably) insecure Hank about his powers, and Mystique comes to Hank's defense with a commendably witty comeback on his behalf. The glance they share afterwards speaks volumes.
Charles helping Erik remember his happiest memory, of lighting the Shabbat candles with his mother as a child, so he could better control his powers. Despite living a much more comfortable and privileged life than Erik, the one beautiful thing that Charles never got to experience is a mother's warm affection. Mrs. Xavier is hinted to be emotionally cold towards her son, so Charles was very sincere when he thanked Erik for allowing him to see and "feel" the cherished memory.
Erik's conversation with Mystique, reassuring her about her true appearence enough to forgo her disguise. He's probably the first person In-Universe who's ever found her blue self beautiful.
Both the American and Russian captains, thinking they're going to die from Erik's missile barrage, tell their respective crews It Has Been an Honor.
The entire beach scene at the end counts; even though he's hell-bent on destruction and revenge, Erik immediately stops caring about redirecting the missiles at the humans when Charles is shot and runs to his friend's side:
Erik: Us turning on each other, it's what they want. I tried to warn you, Charles. I want you by my side. We're brothers, you and I. All of us together, protecting each other. We want the same thing.
Raven running initially to him instead of Magneto in the end. Then he encourages her to go with Erik, because he knows that's what she wants and will make her happy.
When Xavier and Moira kiss at the end, before she promises to never tell anyone where he is, no matter what they do to her. Unfortunately, he still has to wipe her memory. It's heartwarming in a fridge sense, making sure the CIA won't torture her for the location of the X-academy.
After Hank transforms into his Beast form, Erik tells him he's never looked better. Hank grabs him thinking he's being sarcastic but Erik tells him he honestly means it. Havok concurs and gives Hank his new nickname.