Charles and Raven's first meeting as kids, and her running initially to him instead of Magneto in the end.
Also 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:
Another little moment in First Class when the group where just sitting around talking about the powers it made me realize that this is 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. Considering what happens to some of them next..
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 says volumes.
Charles helping Magneto remember his happiest memory, of lighting the Shabbat candles with his mother as a child, so he could better control his powers. It makes their falling out at the end of the film that much more tragic.
Erik's conversation with Mystique, reassuring her about her true appearence enough to forgo her disguise. He's probably the first person who's ever found her blue self beautiful.
The first person in-universe, that is.
Pretty much the entire beach scene at the end counts: Even though he's bent on destruction and revenge, Erik immediately stops caring about redirecting the missiles at the humans when Charles is shot and immediately runs to his side:
Erik: We want the same thing! Charles: I'm sorry, my friend, but we do not.
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.
Which is heartwarming in a fridge sense, making sure the CIA won't torture her for the location of the X-academy.
Pretty much 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!
Both the American and Russian generals, thinking they're going to die from Erik's missile barrage, tell their respective crews It Has Been an Honor.