The moment Ippo wakes up after his very first spar with Miyata to find out that he was accepted at Kamogawa Boxing gym, Sendo continuing to fight even when he was unconscious in response to the cheers of his supporters and Featherweight Champion Date Eiji symbolically handing the baton to Ippo when he retires. Even Coach Kamogawa's backstory can move you to tears.
Just look at the friendship forged between Ippo and the bullies who used to pick on him. At first it looks like they just become fans because as a boxer he must be strong, but as time goes on, not only are they fans, but quite obviously his friends.
Shortly before Ippo's first big non-rookie tournament, his mom fall ill due to overwork. This, of course, leaves Ippo with a huge, heartbreaking choice. And, if things stay the way they are, will force him to quit boxing. Then in comes Umezawa, head of said bullies. He berates Ippo for not asking for his help in the situation, asks to work at the fishing business, and even tells Ippo he quit his other job because "he wanted to follow his heart for once." Then, after a first incredibly hard day (and the realization that all days are like this), one where he almost gave up, he came back the next day anyway. If that doesn't show how true the friendship is, nothing will.
Ippo's relationship with Hiroko, his mother, is a Crowning Moment of Heartwarming ever since episode one. A specific example of this is the conversation Hiroko has with Alexander Volg Zangief after their match, which leaves Ippo comatose for a few days. When Volg asks her if she wasn't worried about Ippo's condition, she says she lost her will to complain after seeing Ippo work so hard.
Hiroko: "Whatever it may be, seeing your son work so hard, is a wonderful thing for any mother."
After suffering his first defeat at the hands of the, at that time, Japan Featherweight Champion Eiji Date, Kimura, Aoki and Takamura head to the locker room to cheer Ippo up. Aoki and Kimura, anticipating that Takamura will be a Jerkass and put Ippo down for his first lost, reprimand Takamura and tell him not to do anything that will worsen the mood. Upon entering the room, everyone holds their breath, expecting Takamura to lay it into Ippo, but he simply pats him on the shoulder and tells him he did a good job.
Turns into a Crowning Moment of Funny when he backpedals after everyone gives him an odd look showing that they weren't expecting that, says exactly what everyone thought he was going to say and is immediately whooped by Coach Kamagawa and thrown out of the locker room.
Volg's return in the manga to Japan to get his Boxing gloves is one giant moment of fluff as Ippo and he get along fantastically, as he stays with Ippo and Hiroko for a few days. He leaves begrudgingly knowing that being there in Japan was "too warm and kind", and that if he didn't leave then, he'd never want to go. It becomes a Tear Jerker when he asks Hiroko to say "I'll pray for your success son" in Russian in his dead Mother's place, and responds "Thank you Mother". He leaves without saying goodbye to Ippo, but knows they already had their own kind of goodbye from their spar, telling Umezawa they'll meet again one day, in the ring.
Shinoda finally getting praised by Kamogawa - especially considering Shinoda feels like his methods of training are dragging Kimura and Aoki down. The poor guy nearly cries and gets all happy that the Coach finally said what he did.
Mashiba's return match after his suspension counts—in a very weird and Mashiba like way. In the year he was suspended, his boss at work put together a Mashiba fan club and cheering section, which Mashiba himself seems to find completely irritating. But, whenever his opponent would sneak in a foul (as tends to happen in his fights), putting him into his old bloodlust, he'd hear their cheers, and would fight back clean. Somehow the final shot of him holding up his belt and angrily yelling "You happy now!?" at his cheering section just brings a smile to the face.