"Hey! Do you know what you just did? You've taken your first step into Kanto. Check your PokéGear Map and see."
"Strong Pokemon. Weak Pokemon. That is only the selfish perception of people. Truly skilled trainers should try to win with their favorites. You know what's important."
After you and Silver have the double battle against Clair and Lance in the remakes, you can go back to Elm's lab and talk to the assistant. He'll tell you that Silver tried to apologize and return his starter, but Elm decided to let him keep it since it had started to like him. Aww.
During that double battle? His Golbat had finally evolved into a Crobat, which is only possible with high friendship. Same principle as the above, still heartwarming to see.
Walking Pokemon. At high happiness, your Pokemon will dance and sing, hug or cuddle you (even the types that have claws - ouch) and find small trinkets like flowers to give to you. It's quite heartwarming when a Legendary, such as Arceus adores you enough when its happiness is high enough.
The moment you arrive in Kanto, and when a more peaceful remix of Vermillion City starts playing, evoking a sense of achievement that you came this far, and nostalgia of coming to a familiar place!
The Olivine City Gym. After the climb through the lighthouse, after being directed to a completely different city for medicine and a completely different Gym, you FINALLY can take on Jasmine. The Trainers in there are some of the same people from the lighthouse; they remember you, respect you, and root for you as you approach their Gym Leader unopposed. And even she thanks you before the battle starts. It's kind of like the end of Episode IV but with a Steelix.
Getting this call on the Pokégear from the Psychic Gym Leader...
"Yeah, this is Sabrina. How are you doing? You.. You'll become even stronger than you are now, I know it... You will defeat every Trainer you face and eventually reach the glowing glory... I just wanted to tell you that. Talk to you later!"
While most Pokegear calls are people bragging about their Pokemon or talking about blunders in catching Pokemon, one trainer calls to talk about how he loves his children and how they're always there to cheer him up, and tells the player to call their mom sometimes, too.
Joey's adoration for his Rattata. It has turned into a bit of a meme, yes, but it's sweet how much this young boy cares for a Pokemon that many others take for granted.
Despite the fact that he's been gone three years, Red's mom still bought him an N64 just in case he came back.
You can meet Janine in the Pokémon League Reception Gate every evening in the remakes. If you do so, she tells you that she just took lunch to her dad, Koga.