Final Fantasy VII is the first story where we come across both Zack and Aerith. It follows the journey of Cloud and company to destroy Meteor and save the world from the likes of Sephiroth. Perhaps the most well-known in the series, Aerith plays a huge role as a character in the game, unlike in Crisis Core. Likewise, Zack plays a small, rather unknown part in this installment. In relation to Aerith and Zack, the timeline in the game takes place after Zack's death, and more than four years after his disappearance in Nibelheim. The dialogue in this particular game is important in that it speaks of Zack and their relationship in retrospect, offering key insights as to what happened to them after.

You first learn about Aerith's relationship with Zack in the game when you, as Cloud, encounter her for the first time in the Midgar slums. Elmyra, Aerith's foster mother, makes an inference about him to Cloud without revealing the entire details. She's not too fond of him either.

Elmyra:That glow in your eyes... you're from SOLDIER, right?
Cloud: Yeah. Rather I used to be...
Elmyra: ...... I don't know how to say this, but... Would you please leave here, tonight? Without telling Aerith.
SOLDIER... the last thing Aerith needs is to get her feelings hurt again...

The SOLDIER she speaks of is Zack, who was, unbeknownst to Cloud at that moment, his former comrade. Her mother had found out about Zack, and his sudden diappearance from Aerith's life. Aerith brings him up momentarily as well, but doesn't say very much either

Aerith: What rank were you?
Cloud: Rank?
Aerith: You know, in SOLDIER.
Cloud: Oh, I was... (pause) First Class.
Aerith: Just the same as him.
Cloud: The same as who?
Aerith: My first boyfriend.
Cloud: You were... serious?
Aerith: No. But I liked him for a while.
Cloud: I probably knew him. What was his name?
Aerith: It doesn't really matter.

Whether or not Aerith is telling the truth, she somewhat plays the relationship off as if a short fling (Or at least the English translation depicts it as so). In actuality, both her and Zack cared very much for one another. She doesn't speak his name, as if to try and forget the sadness. While Zack was captured by Hojo and incapacitated, she spends four years writting him 89 letters, as noted in Crisis Core. When Zack discovers the letters she had written, he also finds out how long he was in a tube. Distraught, he cries out in agony, and sets out to find her in Midgar. Unfortunately, he gets gunned down in the process, unable to see her again. The next time he speaks to her is in the Lifestream, as seen in Maiden Travels the Planet

There isn't any real mention of Zack until you reach Gongaga, where you finally meet his parents. At this point, Cloud has no recollection of him, and only Aerith and Tifa can recall. If you have Aerith and Tifa in your party when you enter their home, you receive the following dialouge:

Zack's Mother: Oh you're right! Don't you know anything about our son?
Zack's Mother: His name is Zack.
Zack's Mother: It's been close to 10 years since he left for the city, saying that he didn't want to live in the country...
Zack's Father: He left saying he's going to join SOLDIER. You ever hear of a Zack in SOLDIER?
Cloud: Hmmm... I don't know.
Aerith: ...Zack...
Zack's Father: Young lady, you know him?
Zack's Mother: I remember he wrote us six or seven years ago saying he had a girlfriend. Could that have been you?
Cloud turns to Aerith
Aerith: That can't...
Aerith leaves the house

Outside
Aerith: What a shock... I didn't know Zack was from this town.
Cloud: You know him?
Aerith: Didn't I tell you? He was my first love.
Cloud: ............
Aerith: Zack... SOLDIER First Class. Same as Cloud.
Cloud: Strange, there aren't that many who make First Class, but I've never heard of him.
Aerith: That's all right. It's all in the past now. I was just worried because I heard he's been missing.
Cloud: Missing?
Aerith: I think it was 5 years ago. He went out on a job, and never came back.
Aerith: He loved women, a real lady's man. He probably found someone else...

In an optional quest, you may revisit Cloud's memories of Zack from when he had abandoned Hojo's lab in Nibelheim. The story has changed throughout the compilation, but this is the scene we are given in the English version of the game.

Zack: What're you gonna do when we get to Midgar?"
(Cloud continues to bob his head.)
Cloud: .....
Zack: I know what I'm gonna do.
Zack: I got a place I can crash for a while...
Zack: No wait, the mother lives there, too...
(He scratches his head.)
Zack: Guess that's out...