The problem with PS1 is that not only does it have a fewer relevant global games not playable on later systems, but many of them didn't age as gracefully as PS2's.

I loved GT1,2, Ace Combat 2, and Medal of Honor, but I haven't played them in ages and looking at videos today doesn't fill me with confidence recommending them. People crucified Jim Ryan for being puzzled as to why anyone would play an old game like classic Gran Turismo, but honestly I can see where he was coming from even if I don't agree. Some PS1 games like Vigilante 8 and Crash Team Racing aged incredibly well, that for my money they pretty much remain the very best in their genres.

My first Ace Combat was the Japanese version of AC3. I was surprised to find out that many prefer it over AC2 which I loved. Maybe I should give it another try one day.

Ratchet & Clank Trilogy, Sly Cooper Trilogy, and Pac-Man World 2+3 come to mind for me. These are games I have yet to get to but I've only ever heard great things about them.

I enjoyed Ace Combat 6 and 7, so I have been curious about playing 4 and 5, but I heard there's no checkpoints in missions so I am a little wary they'd be frustrating to play by modern standards.

There's not all that much that I'm familiar with that you can't also get on more modern systems, usually as some kind of an enhanced version too. But if you like the Dynasty Warriors games, the old games have some good ones too. With the recent remake/remaster, I'm not sure if the original DW3 is worth getting anymore, but DW5 was certainly fun back in the day. And these I haven't played enough to be able to recommend, but the Dynasty Tactics games and the Kessen games supposedly aren't too bad if you like strategy/tactics games, and they're not available on mode modern systems. I've also heard some good things about the Tenchu games. Sadly I have yet to play them, but they might be worth looking into.

I forgot to mention that old ribbon cables will ring scratche a disk and ruin it. So replacing the DVD ribbon cable is a must if you buy any expensive games. 

Also Bob from RetroRGB explains scalers in the top video of this link. https://retrorgb.com/why-retro-consoles-need-a-scaler.html

Valid. Unsurprisingly based on my profile name (unless I just liked a lot of stuff released in 1996), I was born in 1996. The earliest generation as a kid and to the present that I spent a good deal of time on is the sixth generation. Tons of people on this site have a lot of experience with the third through fifth generations, and some even played a good deal of Atari 2600 and so on both growing up and in later years.

Not me. I still have a Nintendo 64 that we got at a yard sale a little after we got our GameCube brand new. That's as old of hardware as I have. And I don't feel much impetus to play it more than I did as a kid. And as for PS1, it seems like most worthwhile games to play on it in terms of quality and sometimes curiosity can just be played digitally as emulation or ports or remasters on PS3 and on. When I have more money one day, I might be interested in getting a PSOne and the screen as long as the screen isn't $400 dollars or so loose by then. But that's purely for novelty's sake. A PS2 plays virtually every PS1 game, and most without issues (even on the revised Slims). 

I like to save older consoles that don't have official repair still available for games that are not as accessible elsewhere. I know there is an appeal to playing games on their original hardware (or even PS1 games on PS2) but I don't feel like wearing down the console playing games I can easily play on other platforms. 

A game I would like to get soon is Soul Edge. It's still quite economical CIB and I've heard it's great. I love Soulcalibur II and VI and enjoy III and IV. I don't give a crap about black labels, so I'd gladly buy it in the Greatest Hits printing to save some money.

Also here is a link that I'm using and others can use for compatibility issues of PS1 and PS2 games across PS2 hardware.

PS1 and PS2 Compatibility - ConsoleMods Wiki

I started with gen 3 (MSX1) around the age of 5 in 1992 (I did occasionally play some Atari 2600 and disliked it, but the Superman game made me jealous!) I didn't even know SNES and MegaDrive existed until late 1993~. As far as games, gen 3 through 6 were all great imo. I really liked Gen 8 and 9 too. Gen 7 (zero experience with the Wii) was a disappointment to me, but you could still find a ton of great games, and the PSP was a truly special hardware.

I adored Soul Edge/Blade! It's my favorite in the series, a lot of it may be nostalgia bias and I'm not sure how well it aged, but I hope you'll enjoy it. Playing this game in the 90's with no internet was a hell of an immersive experience for a fighting game. To this day, the opening remains my all time favourite.