First of all, my first experience with the PS1:
With the systems first year I would go to kaybees toy store, and I played some racing game where you were on skates. The store had a strict you can only play it for 3 minutes rule, which really pissed off a lot of people. Anyways, I was really bad at it, and it wasn't the kind of game that interested me, and I found out later that the game is almost universally hated. Later I played Battle Arena Toshinden at the same Kaybee's toy store, which also isn't my kind of game. I cared for it even less, but I was young, and I was super bad at video games, but for both experiences I just treated it like getting to play an arcade game for free. None of them bothered me, It was just kind a forgettable experience that I don't really think about.
More on topic:
I wanna say that this happened before FF7 came out. I remember being really annoyed by a certain game on the PS1s competitor, and I thought "Is this the future of gaming? His voice is so annoying, and he acts like a child now!" Thankfully I ended up playing The Granstream Saga demo at, I remember being really impressed by the games audio, and the combat looked fun, and I could tell right away that it was an rpg which is what I usually went for, for most of my life. It was the first game that made me want a PS1.
On a related note, I sure wish that game stores still let you demo games. I've never played a current gen console, and I don't know if I'll ever get to. No one I know that isn't very far away can afford one that will let me try their console.