One right from the beginning: Red being in such a rush to see Professor Oak after being caught up in watching a Pokémon battle (the "Gengar VS. Nidorino" one from the Red opening, no less), he falls down the stairs in an incredibly awesome fashion.
And there's his mother's reaction: she looks more exasperated than worried, which might mean it's not the first time Red's tripped over himself rushing down the stairs.
The rapid montages of Gym Leaders imply some of them were on the receiving end of a very one-sided Curb-Stomp Battle, particularly Erika. Blaine's reaction to losing is also priceless.
The Fighting Dojo's leader breaks down as well and, just like in the games, offers to give Red one of his Pokemon, even saying "Choose whichever you like!", which is the same exact message in the text box. Hitmonlee's reaction behind him is priceless.
The look on Blastoise's face when Charizard uses its Megaton Punch, on Blastoise's face.
During Red's catching montage after beating the Champion, we see him attempting to go after a Chansey. As he decides between throwing a rock or bait, it starts to run away, with him giving chase. Everyone who has played the original games will feel his pain all too well.
The same montage shows that Red somehow managed to miss Tentacool of all things on his first trip through Kanto.