Say what you want about Chi-Chi, but her devotion to her family is this. Sure she may be harsh sometimes, but there are a lot of scenes that show that she really does care about them. Such as when she beat up the tutor for insulting Goku and hurting Gohan, her arguments with Bulma over their husbands, and (even though it was a Too Dumb to Live moment) when she slaps Super Buu in the face, because she thought he killed Gohan.
Bulma and Yamcha's relationship. The original series shows that they loved each other dearly despite their differences. They were an Official Couple for the first series and the first arcs of Dragon Ball Z. Even after their break-up, Yamcha and Bulma are still close as Amicable Exes, as seen when Bulma is comforted by Yamcha after she learns of Vegeta's death when fighting Majin Buu.
One from a minor character: Throughout EVERY tournament the gang has been in, there's only been one commentator. Years after, even when the whole world fell for Mr. Satan's lies, the commentator still believed Goku defeated Cell (close enough). Especially heartwarming since he's known Goku since he was a kid, and has been present in a lot of important moments in his life (his marriage proposal, defeating Piccolo Jr. and so many more). Thus, it feels good to see him talk to Goku like an old friend.
Also, early in the Junior Division Tournament of the Buu Saga, the Commentator expresses shock and pride when he recognizes Goten as Goku's son. We even get to see him briefly flash back to Goku and Jackie Chun's fight from back in the early days of Dragonball, which the announcer remembers as the best match he'd ever seen.
The fact that the Pilaf Gang stick together even in their old age is rather heartwarming. One has to think that they stuck around because they formed a genuine friendship over the years.
From the bonus chapter of Jaco the Galactic Patrolman called Dragon Ball Minus. Whether you take it as canon or not compared to the Bardock Special, it is heartwarming seeing Goku's parents going out of their way to save him when they suspected that Frieza was up to something when he recalled all the Saiyans back to their planet. It gives a new light on Goku's arrival on Earth since he wasn't there to destroy it, like everyone told him, but to shelter him from Frieza. It worked, because Frieza didn't know Goku existed until he was right in his face.
What they did also gets more heartwarming when you considered that this action was completely out-of-character for Saiyans since they usually don't care about their off-spring. At the end of the day Frieza was ultimately done in by a selfless act of love by a race not known for it.
Heck, Bardock and Gine in general. Word of Toriyama-sensei is that they were an extreme rarity among their Saiyans because they actually loved each other and in general acted the way a normal married couple would, rather than simply reproducing for the sake of convenience as most other Saiyans did.
Whenever we saw Master Roshi filled with joy because of how far his students have come. Special mention to the last Tournament arc of Dragon Ball where he's filled with pride seeing how far Krillin has come.
Piccolo and Gohan's relationship in general. Even though Piccolo started out as a ridiculously abusive mentor to him, Gohan's innocence mellows him into the biggest HeelFace Turn in the entire franchise, and that's saying something. Afterwards, Piccolo cherishes Gohan like a son, and Gohan looks up to Piccolo as a father figure (almost at the same level as Goku), and their interactions are just adorable. Special mention to the DBZ movies, where Piccolo saves Gohan in almost every single one of them until the Buu Saga. While it might have gotten formulaic, it was still heartwarming to see.
The ending animation of Yo! Son Goku and his Friends Return!! It's basically a long montage of Goku's family: him and his grandfather; him and his wife; him and his wife, son, and father-in-law; him and his wife, eldest son, younger son, daughter-in-law, and daughter-in-law's father. There's a lot of hugs and small moments like Hercule passing food to Goten during a picnic.
Just how well Goku's and Mr. Satan's families meld after their kids get together. In GT, Goku calls him "Grandpa Satan" just like Pan does. In the dub, Hercule refers to Pan as "our granddaughter" when talking to Goku. They're both extremely courteous to each other and respectful of each other's skills. Goku goes out of his way to preserve Mr. Satan's reputation and Mr. Satan goes out of his way to make sure his daughter's in-laws never want. They really do seem to consider each other family.
The title page of the final chapter of the manga is a rendering of young Son Goku and Bulma's first meeting. Toriyama's drawings have clearly evolved over the 9 or so years Dragon Ball started, but it was a neat throwback to where it all began.
The final volume cover has Goku flying in space, waving and telling the reader "Thank you! Good bye!!"
In addition to training them, Master Roshi gave Goku and Krillin reading and math lessons, and basically became their surrogate father. There's also a cute scene where Roshi pauses their training and tells them to take a nap. Goku worries that someone will train and become stronger than them while they do this, but Roshi assures him that everybody needs to sleep, as a montage of warriors around the globe take a nap as well.
For bonus points, Krillin manages to win his match against Chiaotzu with math. Chiaotzu has to stop and count on his fingers to solve 9 - 1 (disrupting his psychic attacks aimed with his hands), but Krillin can add double-digits quickly in his head, because Master Roshi cared enough about him and Goku to make sure they were healthy, well-rounded young men.
Oolong, despite being a huge Jerkass in the Pilaf Saga who disliked Goku, is overjoyed to see him again the day before the 21st Tenkaichi, showing that he's grown to like the kid.
Goku's reunion with his friends, full stop. Bulma even mentions trying to visit Goku by looking for the Kame House, but since Roshi moved it to training Goku and Krillin on a larger island, she couldn't.
Namu and "Jackie Chun" / Master Roshi's interaction after Namu loses his match to Goku. Knowing that Namu entered to use the prize money and buy his drought-stricken village water, "Jackie" gives Namu a container capsule and points him towards a local well, helpfully explaining that outside of the desert, water is free. In return, Namu helps "Jackie" fool Yamcha by posing as Master Roshi in the crowd.
The entire Red Ribbon Army saga is the ultimate heart-warming story of Dragon Ball. It completely embodies the spirit of the original series. It's just Goku as a child wandering the world searching for his lost 4-star Dragon Ball and meeting a lot of new friends and enemies along the way. He rescues #8 and the village elder from General White, he gives away his own heirloom just to resurrect Upa's father, and moved to tears, he meets his grandfather while fighting at Baba's palace.
An easy to miss one during Goku's reunion with his grandpa: Son Gohan, Sr. is fully aware of the Oozaru transformation. He was killed by it. But when he's reunited with Goku there's no anger or resentment towards his grandson whatsoever. He was only concerned that the kid was growing up all right.
This reunion also cushions the blow later when, after seeing Vegeta turn into the Oozaru. Goku finally pieces it together that he was the one who crushed and killed his grandpa all those years ago. While Goku is genuinely remorseful, knowing his grandpa loved him and didn't hold it against him gave him the strength to keep fighting and not dwell on his regret.
Gohan is also very grateful that his old teacher Master Roshi has taken to training Goku, and thanks the rest of the Dragon Ball Gang for being Goku's Found Family.
Episode 94 of the anime has Roshi/Jackie reaching out to Tien, explaining to him he can choose a better path and change himself for the better.
"All your values have been tainted and jaded. It's not your fault, really. The Crane Hermit has always excelled at deception."
The climax of the 22nd Tenka'ichi Budokai in the anime, which led to the fight been taken a few miles outside the arena but ended in Tenshinhan (while completely unconscious) barely winning the match because Goku was unfortunate to land first (he flew in a van), the entire audience vacates the arena and runs towards the two combatants to applaud them. Later, a large man offers to hold Tenshinhan on his shoulders, claiming he must be exhausted from the fight. Keep in mind just hours before, Tenshinhan was very much the biggest Heel seen in a Tenka'ichi Budokai thus far and only changed his ways during the fight with Goku. The same is almost done with Goku, and the two are brought closer together to congratulate each other, and they just smile at each other. Even Krillin was getting teary-eyed.
The end of the episode "Tien's Atonement". During their search for another Dragon Ball, Tien comes face-to-face with an old rival he previously crippled in a fight. After some arguing and even briefly being accused of being responsible for the resent killing of the martial artists. Tien eventually gets on his hands and knees and begs for the ball. Even offering the man he'd crippled a chance to do the same to him. This causes the man to relent, and he gives Tien the ball.
There's something about Goku and Krillin's reunion in the original series episode, "Changes." Krillin had been revived by Goku three years prior but never got a chance to thank him. As soon as he meets Goku again, he runs into Goku and hugs him with tears in his eyes.
His reunion with the gang in general is one, as it dawns on them that the little boy they knew is now a grown man. It's even more emotional in the Japanese version where Masako Nozawa continues portraying him into adulthood, showing that despite his physical growth he's still that boy at heart.
Goku's proposal is also adorable. As a naive little kid, he mistook marriage for a kind of food (understandable, given thisisGoku...), but when he's grown, he nearly breaks Chi-Chi's heart when he tells her so, and then turns around and proposes for real.
Goku: Sometimes my brain doesn't know what my mouth is saying. Good thing my heart does. Will you marry me?
It's a miracle he could disengage his arm from her embrace to fight Ma Junior. Or Tien, for that matter.
The original version just have Goku states that he thought "bride" was something you could eat. "But a promise is a promise, so let's marry."
Chichi is so elated by this it's crazy. When Goku admits he didn't know what marriage was, she immediately tears up, grasping that a promise he didn't understand is not one she can hold him to keep. A lot of the things she says in the next few scenes (and, honestly, the series as whole) shows just how sheltered and naive she is— she still believes Yamucha is madly in love with her, and apologizes profusely to him because she's taken. She cheerfully introduces herself to Goku's friends, talks about what an adorable and smart child they'll raise, tells him she loves him... Chichi is the happiest girl in the world right here. It really informs a lot of their later interactions. Goku's got a bit of a "what did I get myself into" expression on his face, but he cares a lot about making Chichi happy— and for all that she gets mad at him for later, Chichi really admires that good-heartedness to him.
And throughout the fights, of course she's supportive (and worried) about him - especially after his Senzu-bean-induced recovery from a nasty hole in his torso, where he's basically back to full energy/personality. Then when he's cheering ("Wahoo! I did it! I won, yes! I'm the champion! *laughing* WAHOO! YES! *proceeds to do a mid-air jig*), she fondly admits she's never seen him so happy.
After Goku defeats Piccolo Jr., Kami laments that he has not accomplished anything significant during his reign as Guardian of the Earth and he had to rely on mortals to save the day. Master Roshi points out that had Kami not trained Goku, he would never have been strong enough to defeat Piccolo Jr., and if he had never created the Dragon Balls, Bulma would have never gone on a journey and met Goku, met Yamcha, Roshi, etc. If it wasn't for Kami, none of them would have met and become the heroes they are today. Touched, Kami tells Goku that Roshi is a wise man indeed.
Kami: I created this monster...Yet, he's become so powerful as to surpass even me, plunging the world into chaos! I've lost more than power. I've lost my perspective. I no longer deserve to govern this, or any other world.
Roshi: How can you say such things, when it was you who brought our world peace again? If not for your training, Goku would never have attained the level of skill he has now. Junior would have won! And if the Dragon Balls hadn't been forged, none of these young warriors would have ever met. And I can't begin to count the number of people who owe their lives to these kids! When their quest began, they were just children. But they soon grew into heroes. When the world needed saving, eight kids answered the call. One ball; that's how it all began.
Kami: Goku, you're fortunate to have such an excellent teacher.
During the 23rd tournament, Yamucha fights Kami and Kuririn fights Piccolo Jr. Both of them lose their matches, but the crowd still cheers wildly for them— they're old tournament favorites by now, and the audience (including Bulma, Tenshinhan, Goku, Lunch, and Master Roshi) is just so proud of their accomplishments. It's made very clear that their training has paid off (Yamucha debuts one of his signature attacks, a tricky homing ki move called Sokidan, while Kuririn has learned to fly), and they're both humble and grateful for the opportunity to get stronger in the end.
Tenshinhan's victory at the 22nd Tenkaichi Budokai. This entire tournament has put Tien through the wringer: he's here for the sole purpose of beating the Kame School martial artists, but when they find out Goku killed Tao Pai Pai, Tenshinhan's role model and his master's younger brother, the goal becomes to kill Goku. At the same time, he shows a sincere desire to win the tournament and test his strength. He brutalizes Yamcha well beyond necessity for a friendly tournament, breaking his leg, but stops short of killing him, because then he'd be disqualified. Jackie Chun (Master Roshi in disguise) forfeits their match, after talking to Tenshinhan a bit and trying to convince him that he has more potential than just being Tsurusennin's bully. Tenshinhan is stunned by this, because while Tsurusennin's goal is to humiliate Roshi and his students, Roshi's is to encourage his students' improvement by the experience and the reminder that there's always someone stronger. Tenshinhan fights Goku next, and Goku, being Goku, gives him a run for his money. Halfway through, Goku starts shouting at Tenshinhan to stop cheating. Tenshinhan scoffs at this, but eventually he realizes Tsurusennin is forcing Chiaotzu to incapacitate Goku psychichally. Goku apologizes once Tenshinhan demands to be able to fight Goku fairly, because he realizes he wants to test his strength and improve more than he wants revenge or to be an assassin like Tao. Tien is still in shock, because his refusal to kill Goku and insistence on fighting fairly has amounted to a betrayal of his master, but he allows Goku to beat him down for a bit without fighting back, to make up for the blows he landed with Chiaotzu's help. This tournament, he says, is all he has left now. He uses his strongest attack to destroy the ring, warning Goku to dodge it because he doesn't want to kill him, reasoning that if there's no ring and Goku can't fly, he'll lose by ring out as soon as he dodges, since Tenshinhan can fly. Goku manages to use a Kamehameha to propel himself into Tenshinhan and injure him enough that they're both in freefall. Goku crashes into a truck and lands first, losing by ring out. The entire crowd runs over to cheer and congratulate them, Tenshinhan and Goku are hoisted on people's shoulders to greet each other, and Tenshinhan is awed by the positive attention. Master Roshi assures him he's earned it, now that he sees how much better it is to seek to improve yourself than to vanquish enemies. For bonus points, Yamcha is the first one to start the applause for Tenshinhan. Tenshinhan tries to give Goku half the prize money later, and quietly apologizes to Yamcha, once Yamcha congratulates him again. It's an exhaustive HeelFace Turn, and surprisingly emotive, especially if you've come from DBZ first.
Near the beginning of the 23rd World Martial Arts Tournament, Yamcha, Krillin, and Goku ask Master Roshi if he brought them uniforms to wear, like he always used to. Master Roshi says no. He says that because they've all grown up and so vastly surpassed him, they don't have to wear his colors anymore. Cue the reveal that all three of them (separately) saw this coming, and each came prepared with his own custom-made copy of the Turtle Hermit uniform. They honor their teacher not because they have to, but because they choose to.
After King Piccolo's defeat, Bulma is wishing to Shen Long to revive their friends slain by the demons when Yamcha amends her: to revive all their victims.
One among the victims provide another one: it's Man-Wolf, whose human form was so ugly that everybody believed he'd never find a girlfriend... And yet the scene of his resurrection shows he did once he was unstuck from the wolf form that kept him from asking a girl out, and he did it in the few days between the last Tenkaichi and being killed by Tambourine.
Near the beginning of episode 143 "Battle for the Future". The fate of the nerdy-looking human who's body Kami borrowed to enter the tournament. After Kami is sealed thanks to Piccolo reversing the Evil Containment Wave. The man awakens in the ring confused, having no memory of what happened. The poor guy is praised by the crowd, who enjoyed watching him fight, he's even congratulated by his son. It's nice that he at least got a happy ending out of the ordeal.