I got my first Playstation in 1998 (with Abe's Exoddus and Disney's Hercules) but basically everyone around me had one from '96, so I had been able to play well before that.
I'm not sure which game was the first I ever played on a Playstation, my memory isn't sharp enough for that, but it was one of three, at least, so I'll mention them all.
Resident Evil
My cousin David was that cousin who was about ten years older and, to a kid, seems like an adult, but one who's cool and not a total lame-o. My brother especially loved hanging out with him and we would stay over fairly often. Dave had a Playstation and I remember seeing the cover for Resident Evil on his floor (he was fourteen, the floor is a responsible storage solution when you're that age) and already being scared. Why is this big man with a big gun so scared? I figured playing the game would make me cool because I could tell people I played that totally scary game that's rated '15', so I put it on while Dave and my brother were outside playing football and promptly shat myself. Twenty-one years later I still love it.
Rayman
My other main source of a Playstation was my then-best friend, Mike, who was the means by which I played these next two games (I just can't remember which I played first). We were both utterly crap at Rayman and could only get as far as Band Land for a very long time. Rayman is like distilled nostalgia to me today. The soundtrack is so good, and the art style so unique, and seeing or hearing it is like time travel. It makes me feel all cozy and peaceful, in a sad way, as if the world back then was like living in the utopia of your favourite childrens' show. I don't know. I just associate it very strongly with being a kid in the 90s.
Discworld 2: Missing Presumed...!?
I do remember Mike telling me this was the only game he owned for his Playstation that he didn't like and I'm pretty sure that's why I wanted to play it so much. Also the cover was a crazy-looking wizard flying an out-of-control horse in front of a sunset. Mike might have hated it but I loved it, so much so that I promptly stole it from him when I got a Playstation of my own (what? He wasn't ever going to play it).
Without any exaggeration, Discworld 2 is the most important game I've ever played. It had a huge part in shaping my sense of humour (even introducing me to Monty Python), enamouring me with Fantasy as a genre, and, not least of all, introducing me to Discworld. Because of this game, I bought the books. Because I bought the books, I became an avid reader. Because I became an avid reader, I wanted to become a writer. I studied writing in university because of this game. I talk and joke the way I do because of this game. I love genre fiction because of this game. It's shaped me as a person more than almost anything else. I'd like to say it was my first Playstation game, since that would be very narratively satisfying, but I might have played Resi or Rayman before it.
Tell me this music doesn't make you want to go on a mysterious magical adventure: