My most vivid memory from my first playthrough was Aeris's Death - because my mother completely ruined it for me.
Like a lot of parents she was always tolerant of mine and my little sister's geekery, but never really shared in it; we were
obsessed with this game, so she knew all of the characters' names and their relationships and some of the story, but she had no emotional attachment to them whatsoever.
So Aeris dies. Me and little sister are all O_O at the screen. Mom starts cracking jokes.
Most of these were related to Tifa and Aeris's rivalry over Cloud, but the one that really made us bust a gut was when Cloud first picks Aeris up - Mom says "C'MON, DUMP HER IN THE WATER, CLOUD!!"
And then he does.
Needless to say, the emotional resonance was kind of lost (though later replays of the scene produced the appropriate amount of sad).
Other things...
I hated Cid. And Barret. And Red XIII. The latter two I think were because of their respective sidequests, but I have no idea what turned me off to Cid.
I had to use a walkthrough to find out where to go after I acquired the Tiny Bronco, and for some reason remember the walkthrough saying TIFA died at the end of Disc One...so for some time I thought the date mechanics also affected who Sephiroth killed.
When I encountered Sephiroth in the Nibelheim basement in that one optional scene, it caught me completely by surprise, and I was sure he was going to kill me. When he threw the Destruct materia, I was so keyed-up I actually screamed. My sister still teases me about this.
Like a few other people in this thread, I also thought Sephiroth was supposed to join you at some point. I actually didn't realize until he killed Aeris that he was going to be the endgame villain - after all, AVALANCHE was gunning for the same people he was, and they were the Good Guys. I thought he was basically a more violent version of Aeris, a Cetra who'd been abused and imprisoned by Shinra. I spent the escape from the Shinra building onward thinking that he was sort of a "mentor," that he'd tell you all the plot twists and your True Mission once you caught up to him. ...which turned out to be correct, though RATHER DIFFERENT from what I was imagining.
Oh, and I had a moment of zen the first time I ever heard Cloud's Theme on the worldmap, where all of my thoughts sort of stopped and I just let the character stand there while I listened. I think that moment was what cemented this game in my Favorites list forever.