In the first episode, the original Digi Destined's partners are getting brutally attacked by the Digimon Emperor, and Agumon cries out for Tai. What does Tai do? Drop everything and head to the Digital World to help his friend.
Matt, Joe, and Sora do the same in later episodes.
In "If I had a Tail Hammer", Minomon's (Wormmon's pre-digivolution form) energy and enthusiasm while suggesting Ken exercise his excess dinner off is both funny and heartwarming. Because it shows Wormmon has since recovered from his past woobie status. The overall mood set by his and Ken's exchange of dialogue at this point shows their friendship as a whole has also recovered from the damage caused by Ken's role as the Digimon Emperor.
When the chips are down and it looks like the gang are going to be bug chow (it makes sense in context), Davis gives Ken a big speech about the Power of Friendship which activates their Digimon's power to combine.
Even their lines in the dub manage to keep it heartwarming.
Davis: Cool! I can hear my heart beating. Wait, that's Ken's heart.
Ken: So this is what it feels like to have a friend. ...It's not half bad.
When Ken woke up from his self induced coma he went back to the Digital World. After recovering from a case of Laser-Guided Amnesia, and being beat up by some baby Digimon. He has a total meltdown when he finally finds his partner Wormmon.
Cody accepting an invitation to Ken's Christmas Party. Look, everyone! They're on speaking terms!
Cody: I'm glad I made your list!
This troper found the end of episode 13 a Crowning Moment of Heartwarming when TK assured Kari, who had been on the verge of suffering from a Heroic BSOD due to the Dark Ocean playing on her inferiority complex regarding her brother, that she didn't just have Tai to lean on—she had him and the rest of her friends too.
Though it's slightly sweeter in the dub, where it's implied the two may have feeling for each-other, though the epilogue shows nothing may have came out of it.
Ken's mom gets one of these too. Once she and her husband are bluntly made aware of their considerable screw ups concerning their treatment of their youngest child, the audience gets to see her as a mother who truly loves her son, and we start to realize how hard she's trying to make up for her mistakes. Anyway, after countless episodes of Ken's beating himself up, trying to atone alone, forgetting who she was, and generally going through so much crap that it became difficult to imagine he was ever the Kaiser, he finally comes round to the others attempts to help him. During one episode, Ken calls Mrs. Ichijouji to tell her he's staying at a friend's place, and she starts crying on the phone because the son she'd thought was trapped in a bubble forever and would never be able to deal with the social world finally has friends.
Chibimon and Minomon waking Davis up by playfighting on his bed is just So. Damn. Cute. You'll nearly explode watching it.
Ken's laughing (as in not a maniacal evil laugh, but a genuinely happy one) for the first time (much to the others' astonishment) got This Troper, too.
Did anyone else go into full Squee mode at how adorable that was?
Davis's speech while Ken is trying to close a gateway to the Dark Ocean (and is in considerable pain while doing so) could be seen as heartwarming. The speech, plus the fact that every single one of the team is right there next to him helping him stay upright and encouraging him, is pretty damn moving even when the Dub does it.
This Troper found the uniting power of an entire planet's worth of Chosen Children right at the end of the second series tugged at the heartstrings quite a bit.
This troper as well. A lot of people deride that scene, but it's amazing when you realize just how well-set-up that was- the only requirement to be made a Digidestined is to see a Digimon, and it's likely that, just like the new protagonists of 02, all those kids were supporting Tai and company against Diaboromon. Every single one. And then they get repaid in full by becoming chosen kids themselves and taking an active role in the final battle.
The whole series finale is made of this, even if you don't particularly like some of the last minute hookups it's great. Davis leading the charge against the Big Bad, with the power of wishing? Love it!
Not to mention every single person on earth gets their own Digimon (Seriously every human has their own personal best buddy for life. How awesome is that) Kari earlier in the season fantasized of children and Digimon playing in harmony, which at the end comes true. And the Digital World even gets respected enough to be part of the United Nations! Seriously this is probably not just the most feel good ending of all of Digimon, but any show period.
All this with "Bokura no Digital World" playing the background, a song where all the main characters thank the Digital World in their own little way for everything they've done and grown up to become, joining together in the choruses. Maeda Ai's solo rerecording is just as tingly.
When the kids have had a rough day and are ready for Joe's brother to take them home, TK gives up his seat in the car for Ken. This small act of kindness shows us that the group has finally accepted him, and maybe by taking up the offer he's starting to accept himself.
Ken and Wormmon's song, Honto no Tsuyosa. Ken and Wormmon reflect on how they've been through so much together (the lyrics imply this takes place post Ken's Heel-Face Turn), and Wormmon's Undying Loyalty even in the face of Ken's cruelty.