It's established that Quicksilver hates being still, because of his accelerated sense of time. Despite this, he sits down to hold his little sister and watch TV with her.
Watch Erik carefully in the scene where Peter is rescuing him. You can see that the man really wants to be annoyed with the younger boy because of his attitude, but is restraining himself because A) Said boy is rescuing him, B) Said boy is also a mutant, therefore a brother in arms, and C) depending on how you interperet Peter's Mythology Gag, he may or may not be his son.
The ending, where Wolverine wakes up in the changed future.
He's walking around like he can't believe he's in such a cheerful and familiar place. Think about it; he spent decades in the Bad Future and now he's back in the closest thing he has to home.
He's overjoyed to see Jean alive again and even Scott too. "It's good to see you, Scott," is heartfelt considering their adversity in the first two films.
Just seeing the school bright and whole and full of children again.
Jean: Logan, is everything alright? Logan: ...yeah. Yeah, I think it is.
Early in the film Wolverine tells Beast "You don't know this, but you and I are gonna be really good friends." Even in the changed future this is completely true.
Logan in the righted timeline is a faculty member at the Xavier school; the history instructor. He truly found a place to belong.
Wolverine returning from the past to the peaceful, happier timeline of the present and greeting the Professor after seeing Scott and Jean. The way the Professor's face lights up when he realizes that this is the Logan that helped him in the past without any recollection of the last fifty years is full of relief and comfort for both Logan and the audience and a confirmation that Logan's hard work to fix things in the past paid off.
A slightly meta example, but hearing John Ottman's X-Men theme for the first time in 11 years brought a smile to the faces of several fans.
Future!Professor X briefly shaking Future!Magneto's hand while the latter is dying. Plus Erik lamenting that they spent so many years fighting each other when they should have been working together.
This simple gesture, while brief due to the Sentinels about to barge in, shows that whatever difference these two had, they were still friends. And now they have finally buried the hatchet after all these long years. It's sad too, because they know there's nothing else for them since they'll both die soon, but at least they have finally made peace.
Magneto: All those years we wasted fighting each other, Charles. To have a precious few of them back...
Wolverine trying to comfort Charles after the latter's meltdown in Cerebro.
Also when he asks Charles to find him and all of the future X-Men and help guide them. It's a really sweet moment from a guy we're used to seeing act grumpy and cynical most of the time.
Also the fact that Wolverine is subtly admitting that his time with the X-Men has been the best times of his life.
Meta example: both Patrick Stewart and Ian McKellen were surprised when Bryan Singer approached them directly about the movie (during a theater production of Waiting For Godot that the two were appearing in) as they thought the roles of Xavier and Magneto had been passed to James McAvoy and Michael Fassbender, and that they themselves would never play the roles again. According to McKellen himself, when he and Stewart learned otherwise, they were delighted and signed on right away, eager to play the parts again and to work with the younger cast.
Charles convincing Raven not to shoot Trask by appealing to her better nature. Despite that he could telepathically force her not to do it, Charles lets her make her own choices, stating that he has faith in her. And it works.
Before that, Mystique disguised as President Nixon offering herself to be killed by Magneto, in place of Nixon and his entire Cabinet. Then saving them all. Even though she has every reason to hate Nixon's administration for approving the Sentinel programme.
Before she leaves, Raven looks at Charles and Hank and smiles warmly.
Nixon cancelling the Sentinel program, and potentially pardoning Mystique.
Future!Charles talking to past!Charles and showing him all the good he can do, giving him hope again. And he finally starts acting like the Xavier we all knew.
Past!Erik asking Charles for a game of chess after they've had a huge argument, and Charles eventually agreeing. Plus, the way they almost casually say goodbye to each other when Charles lets Erik go after he's tried to kill the president, starting their strange Friendly Enemy relationship they'll have most of their lives.
After Future!Erik is hit by shrapnel and is dying he does not fight the Sentinels but rather strengthens the Barricade. He uses the last of his strength not to fight his enemy like he's been doing his whole life, but to protect his friend.
Right before they part ways at the end of the film, Magneto leaves his helmet behind on the ground. Yes, Mystique may have pulled it off his head in an earlier scene, but Erik looked right at the helmet — which was directly next to his feet at the time — and continued on without it. Considering Erik's obsessive need of it in most other parts of the series, this speaks volumes about his trust for Charles, even if he only left it behind out of self-preservation.
On that note, Future!Erik doesn't wear it, either.
Although the younger Charles hasn't seen Mystique in 11 years by the time he finds her at the Paris hotel, his first instinct after she is stunned by a Taser is to try to comfort her by caressing her hair and shoulder while telling her that he will keep her safe along with other soothing words. He also stands in between the barrel and his adoptive sister after Magneto points a gun in her direction, and Erik later has to use his power to change the bullet's trajectory so that it won't hit Charles in the head. Despite their estrangement, Xavier is willing to die for Raven to save her life.
And to add to this, Mystique seems genuinely happy to see Charles again when they first meet up.
Later, when Mystique is getting her leg bandaged, the nurse wonders if the mutant girl from the TV footage has a family. Mystique (in disguise) wistfully replies "Yes. She does."
In a bizarre way, Beast beating the heck out of Magneto and doing his best to drown him for trying to kill Raven. Hank still cares deeply about her.
Havok attempting to talk down Mystique from killing Stryker. She may have abandoned the X-Men, but you can tell he's glad to see her and doesn't want her to pull the trigger.
Not to mention that when leaving, he pauses and asks, with a small amount of audible sadness, if she is going to leave with them. When she declines and goes to continue her plans, you can tell that Havok misses his old friend.
Towards the climatic battle in the past towards the end of the movie, Hank asks Logan whether or not he makes it in the future. Logan truthfully replies that he doesn't to which Hank gives him a startled look, but Logan then kindly reassures him "But we have a chance to change that." It really encompasses the theme of hope in the movie.
In the very beginning of the film before the title credits, we see a young mutant boy with a streak of white in his hair scavenging among the ruins of the decaying city shortly before a Sentinel catches up. It's a blink-and-you'll miss it sort of moment, but after Logan wakes up in the Good!Future, we see that same little boy walk past his door in the Xavier school, healthy, seemingly happy, and alive.
For certain shippers, seeing Bobby and Rogue together again at the end.
With the film's explanation that the blue furry form for Beast is him hulking out when he loses control of himself, him being that way all the time in the future could be interpreted as Beast no longer feeling like he needs to hide that he is a mutant.
Mystique saving four mutants, including Havok and Toad from the horrible fate of being used as lab rats shows she's the only one fighting for Mutant rights.
A human Senator pointing out to Trask that if mutants exist, then they are living among humans peacefully. This guy in the 1970's is more tolerant than most of the humans we meet in the 21st century.
Logan telling Xavier "I was your most helpless student once." It reminds us that in the first movie, when Logan was bitter and alone, Xavier was the one who guided him into both being a better person, and having a sense of purpose and belonging. Now the roles have reversed, and Logan is guiding the bitter Xavier into becoming a better man.
Logan: I don't know how long I've got here, but I do know that a long time, actually, a long time from now, I was your most helpless student, and you unlocked my mind. You showed me what I was, you showed me what I can be.
One example that is mixed with Tearjerker is the part of the narration that explained the fact that when the Sentinels started their genocide of mutants, many humans stood up for mutantkind. While it did not change the outcome, when it came down to the wire, a chunk of humanity tried to do the right thing and did not simply abandon the mutants to their fate. Charles was right all along.