The date in the Golden Saucer gondola to the backdrop of "Interrupted by Fireworks", with either Tifa or Aerith. The date with Yuffie uses the same music, but it's Played for Laughs, and the date with Barret uses Barret's theme and is utterly hilarious.
Another scene, same music: Before leaving the Temple of the Ancients when Cait Sith fortune-tells Cloud and Aerith about their future prospects and then sacrifices himself to help Cloud to get the Black Materia.
Everytime Barret talks about Marlene and proves that he, despite being so hot-tempered and even violent, has a soft side, too.
Elmyra: "You're her father!? How in the world could you ever leave a child alone like that!?"
Barret: "...please don't start with that. I think about it all the time. What would happen to Marlene, if I... But you gotta understand somethin'... I don't got an answer. I wanna be with Marlene... But I gotta fight. 'Cause if I don't…the planet's gonna die. So I'm gonna keep fightin'! But, I'm worried 'bout Marlene. I really just wanna be with her... always. See? I'm goin' in circles, now."
The scene with Cloud and Tifa underneath the Highwind the night before they go into the final battle with Sephiroth. Even if you hate the Cloud/Tifa ship, it's really just sweet when they both realize that they're the only thing the other one has left... Kind of sad too, but still.
Afterward, when Tifa comments on how the Highwind's too big for just two people and how lonely it is, Cloud tells her not to worry, starts doing squats, and say he'll "make a big enough ruckus for everyone". Then of course everyone shows up just afterward.
Yuffie, despite coming off as an apathetic Jerkass in the past, is actually probably the most broken up over Aerith's death. She actually breaks down in tears in Cloud's arms. Rather touching, and Yuffie noticeably gets better afterwards.
Something about Tifa's desperate caring for Cloud in the end sequence when he is on a rock ledge, the whole cavern is falling in, and Tifa is calling and reaching out to try to get him to safety. Followed shortly by another example when the Highwind comes to the rescue (also equals as a Moment of Awesome, of course).
Cave of the Gi. You know the scene
"My father was a coward!" "Actually your father is very likely the only reason we're still standing here today."
"I am Nanaki, son of Seto, warrior and protector of Cosmo Canyon!"
In Elmyra's flashback to how she met Aeris, there are two NPC couples who reunite after being away during Shinra's war with Wutai: One young man returns to his young significant other and twirls her about in the air in jubilation. Then an older, paunchier man steps off the train, and his significant other does the same thing to him!
Cid finally getting to go to space, and finding out (the hard way) that Sherra was right about the oxygen tank the whole time. Shares a room with moments of Awesome and Tearjerking.
If you think of Advent Children as a extra long bonus cutscene after the game's credits then consider:
When Cloud's cell is sinking in the forest pool to the tune of that hauntingly beautiful score, it plays this touching message from Aeris: "I never blamed you. Not once. You came for me. That is all that matters."
When the whole gang shows up to fight Bahamut.
When Cloud has reawakened, the kids are gathering around him, and he's looking embarrassed by all the attention. There is something about the strong SOLDIER and the group of adoring little kids. And he's smiling a little and looking kind of awkward and embarrassed by all the attention, but he looks so happy. And then there's the last two lines of the movie, set against that, with Cloud actually smiling. "I'm not alone."
You don't really see Cloud get embarrassed until in the the middle of all that he spots Tifa. And she gives him a very reassuring smile.
"There is not a thing I don't cherish."
"This is where a hero began his journey." What? No, it's just...you know, something in my eye, that's all...
Priscilla. You save her life early on, and if you check up on her you find out she's developed a crush on Cloud. After Cloud goes catatonic and then comes back you can still visit her. She gets mad because "You have to take better care of yourself. We're going to get married some day!" And Cloud rubs his head and apologizes. So cute!
YMMV, but in Red XIII/Nanaki's story in On a Way to a Smile, Red XIII is having doubts about himself due to his beast nature fighting with his human like traits, Vincent of all people explains that it's due to his long lived nature and fear of loss, Vincent comforts him and promises that the two of them will meet up once every year for the rest of Red XIII's life, so Red XIII can tell him all about his life and so he won't be alone.