First, Cloud floating near death and being healed by Aerith, with her and Zack joking about him being too big to adopt after Cloud calls her Mother. Even when they're dead, they're still looking out for Cloud. Cloud then wakes up in the Sector 5 Church with all of the people there.
The look on Cloud's face - embarrassed, amazed, happy as the kids crowded around to celebrate - and then when Aerith turns and walks away, and then Zack waves once and follows... and then Cloud looks at the kids again, and his friends, and Tifa, and the true moment is the line after the fade: "I'm not alone. Not anymore." Just...
Heck, just the fact that his embarrassed smile is followed by a genuine, bigger smile for the first time in the movie as he sees Aerith and she reassures him before leaving is both heartwarming and tearjerking.
Sephiroth: Tell me what you cherish most. Give me the pleasure of taking it away. Cloud: I pity you. You just don't get it at all. There's not a thing I don't cherish! Commence Omnislash ver. 5
In Advent Children Complete after Sephiroth has hit Cloud with an Octoslash and done some damage, Zack shows up to offer some encouragement and support.
Zack: So what if it looks hopeless? If it was me, I still wouldn't give up. Embrace your dreams, and no matter what happens, protect your honor as SOLDIER. Well, okay, you never made SOLDIER, but it's what's in here [Puts fist over heart] that counts. Cloud: Zack... Zack: So, need any help? Cloud: No. [Cloud starts to stand up] Zack: You already beat him once didn't you? This should be a cinch! Cloud: Yeah. Zack: Cloud, you know what I told you. Cloud: That's right, I am your living legacy. ''Commence Omnislash ver. 6
A small one, but when Marlene is angry at Cloud, she runs to the only other person there: Vincent. Vincent looks surprised, but he still lets her hide under his cape until Cloud has stopped being a jerk.
In the ending scene in the church, Denzel's Geostigma is cured. As Yuffie jumps excitedly up and down, if you look closely you can see Vincent give the tiniest of smiles at the whole scene.
When Cloud drops his phone into a pool of water, we watch it slowly sink, as we hear messages from all his friends. Then, when it finally reaches the bottom:
Aerith: I never blamed you. Not once. You came for me. That's all that matters.
In AC Reunion Files, the creators originally considered having the last voicemail be a really old one from Aerith when she was alive. They changed it since they thought it would be too spooky, but the idea that Cloud would keep a message this long is both tearjerking and heartwarming. Not only that, but the fact that he couldn't bring himself to delete the messages shows his love for the gang, and the gang similarly overflow with love and concern for him in their voicemails. Particularly cute is Reeve laughing and saying 'You never change, Cloud', followed by 'I was thinking of helping you out, if that's fine by you.'
Cloud saving Tifa from the falling building. Not really the moment itself but if you think back to the original game and Cloud's promise. Yep she finally got her hero helping her out when she's in a bind.
In the special OVA that comes with the Complete version, Denzel tells Reeve the story of his life leading up to getting found by Cloud. One segment is about him being taken in by an old woman after his parents were killed in the Sector 7 crash. At the end, when Reeve is leaving, he tells Denzel, "Thank you for taking care of my mother."
In the Complete version, there's a scene where Marlene does an adorable imitation of Barret's tone and mannerisms, and refers to him as her father.
Later in the same scene:
Cloud: In the past youíre always looked after me. You and Denzel and Tifa. You were there whenever I needed you. Well now itís my turn.
Rufus leaping off a building to escape Kadaj with no means to save himself from the deadly fall. How can he do that so readily and coolly? According to Word of God, he's just that confident that the Turks will save him.
When Reno and Rude see Bahamut Sin, they freak out and run, but stop to pick up three kids that are under the white-haired trios' control (as they are so entranced that they don't notice the danger).
Kadaj mistaking Aerith for his mother, enabling him to pass on into the Lifestream. The dialogue between Aerith and Zack in the Lifestream is also heartwarming.
In Complete, after the credits, Denzel is taken to Zack's resting place, but the Buster Sword is missing from the grave. In it's place is a bed of flowers in an otherwise barren wasteland where Midgar lie in the distance and then we see the sword, once broken and rusted, restored to it's once-beautiful, pristine condition as it lies in a new resting place: inside Aerith's church.
Denzel: "Is this somebody's grave?"
Cloud: "No. This is where a hero began his journey."