The flashback. Yes, that flashback. The one where kid Kai gives kid Aichi Blaster Blade. To clarify, Aichi's been beaten up half-to-hell, walking along bruised and battered, and Kai comes from pretty much nowhere. And gives him Blaster Blade, which in canon is an uber-rare hard-to-get card. He proceeds to give Aichi a pep talk, telling him to imagine himself as strong as the warrior depicted on the card, and that one day, he would be as strong as Blaster Blade. While Aichi isn't there yet, Kai's gift and the pep talk have pretty much started him on his way, added to by his taking-up of Vanguard.
A short, but sweet moment in episode 13: As the final match in the Card Shop Tournament between Kai and Kamui begins, Morikawa notices Aichi still upset about his loss to Kamui. He and Izaki go to cheer him up and watch the match with them
This combined with everyone's reaction to Kai, Kamui, Misaki, and Aichi teaming up to join the Vanguard National Championship gives the last few minutes of the episode a slight Tastes Like Diabetes feeling.
Misaki has every reason to pull out of Vanguard fights considering what happened to her in the past and how the fights bring back those memories against her will. The heartwarming part is when Shin helps her remember that Vanguard fights also bring back kind and warm memories of the past as well, pulling her out of her depression.
And then there's the flash backs of Misaki's parents in the same episode, it's just hard not to smile when you see them
Episode 64. Aichi's soft and cheerful tone makes this one line particularly touching.
Psyqualia!Aichi: "Long time no see, Blaster Dark. It's finally time for your attack."
Aichi's last use of the phrase "Picture it" during his finishing move against Ren. The following speech is incredibly heartwarming, especially in contrast to how the phrase had been used until that point during the PSY Qualia-filled match.
The understated, but definitely present reconciliation between Ren and Kai.
At the very end of the episode, Kai finally acknowledges that Aichi truly has become strong, and challenges him to a fight.
Episode 84. Ren approaching cooking the same way he approaches Vanguard, with Asaka cheering him on. Seeing it in action is absolutely adorable.
Episode 85. Aichi's bonding with the Gold Paladins, finally accepting them as his new comrades, not as a mean to get his Royal Paladins back.
Episode 86. Gouki, out of all people, is the first to realize that Kenji has a problem, and outright gave his advice in the middle of the fight, in a pretty hammy way. Doubles as Moment of Awesome.
Episode 106. Naoki is willing to help Aichi after not doing anything in the past.
Episode 109 had Misaki coming to Aichi's aid out of her own respect and friendship to help Aichi with the club, despite Kourin's machinations to prevent her and wanting to snap the fellow student awake. That's loyalty for you.
Every single scene that contains either Misaki and/or Kourin in it during the school play. From Kourin willing to run all the way to school from a shoot to be there for her friends, to Misaki calling out encouragements during Kourin's fight with Daigo, the whole thing is one big moment of heartwarming.
In Episode 162, with Kai sounding as though he has been Driven to Suicide and blaming himself for the terrible things that have happened, Aichi continues to insist that Kai is responsible for him becoming the person he is, and that Kai is not alone.
Aichi pulling Kai back from the Despair Event Horizon in the finale of Link Joker was pretty heartwarming, too. His sheer devotion to Kai and the way he accepts him and his flaws no matter what Kai sees himself as really hammers in both Aichi's strength and how strong a bond the two have.
Kai trying to cheer up Misaki in Episode 166 definitely counts. Upon hearing her worries about whether they'll be able to find Aichi or not, Kai tells her not to worry about the future and focus on the present, even at one point using a Vanguard analogy to prove his point further. His whole dialogue is positively Adorkable, and shows how much Kai has changed since the beginning of the series.
Let's be honest, most of the interaction between Aichi and Kourin, especially in later seasons due to the plot point of Kourin always in danger of losing her memories. She doesn't know how long she'll be able to stay with him and their friends but she'll be damned if it means she'll become any less devoted to Aichi's safety and happiness. Taken to its limit during Legion Mate. Aichi is now the last connection she has to keep her memories. Aichi is in absolute pain from holding Link Joker in and is ready to seal himself off, protected and known only by strangers. Suddenly Kourin teleports to him and she says that she will share his agony and devote herself to him. You can tell that Aichi is happy to have at least someone that he knows and is close to, be with him during this trying time.