I'm a little bit conflicted about posting here because I LOVED my PlayStation at the time but now there's maybe a handful of PS1 games that I like to go back to, none of which require me to dig out a PS1 to play.
We got a N64 at launch as an early joint Bday gift for me and my brother (they're less than a month apart). We loved it and bought a bunch of games for it (and got some for Xmas). While we never considered getting rid of it, the N64's game droughts were REAL. Having said that, we had associated the PS1 with the winter of '96 in which Toys R Us had a bunch of kiosks of all three systems and Crash 1 was almost always frozen on every single one of the PS1 kiosks because all of their systems were constantly overheating (not sure if that was an issue Sony was still dealing with or if our store just had a bunch of old systems). Because of that and stuff like loading times we felt really good about our purchase.
Eventually, my brother got a job and he wound up buying a PS1 right around the time Crash 2 came out. Prior to him buying a PS1 I had no idea he planned to buy one and was actually playing one for the first time at a friends house when my mom called me saying it was time to go home and she told me that my bro had bought a PS1. I remember the demo disc having Parrapa and some terrible F1 game on it. My bro bought Toshinden. The games that were big deals for me at the time were the Wipeout games, RE1, Crash 2, MGS1, Tenchu 1, Tekken 1-3, WWF Warzone, WWF Attitude, WWF Smackdown 1&2, Monster Rancher (I would mostly just make monster with CDs), Twisted Metal 1, Jet Moto 1&2, Gex: Enter the Gecko, Mortal Kombat Trilogy, MegaMan Legends 1, and that's all I can think of for the moment. I definitely played more both back then and later, but those were the PS1 games I really spent a lot of time with. Having said that, the PS1 never turned us away from Nintendo (or Sega as we got a Dreamcast on 9-9-99), but getting the PS1 was the first time we owned more than one game system at a time.