An early chapter has Team Rocket's Meowth learn Pay Day from a female ninja Meowth. In a later chapter, we learn that this female ninja Meowth is Meowzie, who is the reason why Team Rocket's Meowth learned to act human.
Team Rocket acting competently. I'm not quite sure why, but given how much of underdogs they were, it's really nice to see them get victories, see them actually on Ash's level, get acknowledged by the Boss as being hard workers...
Chapter 20 has the birth of Damos, the first new Pokemon that Professor Oak has gotten for himself in a decade, whom he welcomes into the world, and his home, in a particularly touching manner.
Mewtwo's reaction to getting his memories back. It really shows how much Ash got to the guy in canon.
Ash got most of his badges the first time around in the tests of Character. It's kind of heartwarming to see the guy honestly winning matches, instead of, say, getting rewarded for saving the gym from Team Rocket.
Ash learning that his father didn't abandon him by choice.
In the second part of the Power of One arc, Ash is ambushed by Hunter J, who orders her Drapion to use Toxic with the intention of killing Ash. Riolu, realising this, throws himself in the way to take the attack for his trainer and friend. He subsequently evolves. When Ash realises this, his new Lucario says the following:
Lucario: As you kept faith with me, so I with you. Go. I'll handle J.
In Johto, when Brock helps the baby Stantler again. Large portions of this arc were written from Stantler's point of view, and the heartwarming factor is almost too much to bear. It's crowning moment is when Stantler chooses to go with Brock rather than stay with the herd.
When Ash visits the school in Violet City, one of the students asks him which of his Pokémon he thinks is the best. Ash, who does not and has never played favourites, lists the individual strengths of each of his Pokémon, and then tells Charizard's story without mentioning that it's about Charizard until the very end. Given that the kids all know how awesome Charizard is, Ash's point about not disregarding any Pokémon as weak and the importance of befriending them is well and truly made.