After Ash accidentally destroys Mewtwo's cloning machine and frees Pikachu, Bulbasaur, Squirtle, Charizard and the other Pokémon. The first thing Pikachu, Bulbasaur and Squirtle do when freed is give Ash a hug.
Especially notable for Bulbasaur, who started off cynical to Ash originally, convinced a human would abandon him in a flash. After being freed, Bulbasaur awakens very groggy, but the moment he hears Ash's voice he bolts up and runs up to him overjoyed.
When Mewtwo starts nabbing everyone's Pokemon, we get two from Misty; the first is her hiding Togepi in her bag (making Togepi the only one who isn't captured) and the second is her angrily demanding that Mewtwo give Psyduck back after he's taken away.
The moment when Mew swoops in out of nowhere to save Ash just before Mewtwo nearly killed him. He didn't know Ash very well by that point, and he didn't have to save his life. But he did it.
The dub Mew gives a good speech about how wrath doesn't get you anywhere.
Mew: (Meowth translates) You don't prove anything by showing off a lot of special powers, a Pokemon's real strength comes from the heart.
The scene where Ash gets turned to stone, and poor Pikachu keeps trying to wake him up started out very sad. It became a Moment of Awesome when Pikachu's tears and not its electricity brought him back.
"I see now that the Circumstances of one's birth are irrelevant. It is what you do with the gift of life that determines who you are."
Even more so in the Japanese version, where Mewtwo's Heel Realization is connected to the plot/message that's been discussed and explored throughout the movie. He finally realizes that it's pointless to prove that his race of clones is superior through force, loses the ego that he's been trying to feed throughout the movie, and decides that life is just worth living to the fullest, coexisting peacefully with other humans and Pokémon.
The last lines of the movie make a great Bookend to the series in general. After Ash catches a glimpse of Mew he reminisces about his first steps.
Ash: Hey what's that?
Misty: What's what? I don't see anything.
Brock: Except the sky.
Ash: The day I left home to start my journey, I saw a really rare pokemon. And just now I thought I saw another one.
Misty: Well maybe you're seeing things Ash.
Brock: Well maybe he's not.
Ash: Well maybe we'll see.
While a minor funny moment, before Meowth reveals that Jessie is a Lethal Chef, he at least thanks her before hand. One of the rare moments where a fellow member of Team Rocket cares about one and other. In this case Jessie offering to cook for a starving Meowth, and Meowth in turn turning her down politely.
In "Pikachu's Vacation", Pikachu trying to convince the bully Pokemon to help get Charizard's head out of a pipe (long story). It's also implied that the reason Pikachu is trying to save Charizard (who's usually an uncooperative jerk) in the first place is because he still sees him as a friend.