All the Crash Bandicoot games, Digimon World (the first one at least), Tony Hawk Pro Skater . And I also like the Tazan game too

Digimon World 1, 2 and 3. And Digimon Card Battle.

FFVII, FFVIII, FF9

Spyro trilogy

Many more but good Start here.

Also, if you like RPGs you should try Legend of Dragoon and Vagrant Story, two great games that are still stuck on PS1.

The Crash Bandicoot and Spyro trilogies are essential.
The Wipeout games are also great.

Original hardware/games are the way to go in my opinion for that authentic feel, though the problem is you'll need a decent adaptor for a modern TV.

Suikoden/Suikoden 2: Both excellent JRPG game. If only going to play one guess I would choose Suikoden 2 since it took what was good with Suikoden and made it better more polished.

Alundra - Zelda: Link of the past style game but with a much darker theme and way harder puzzles

Any 2D fighting game is the worst version on PS1 and is better played on Saturn. 

Heart of Darkness is a great adventure platformer that I keep thinking came out on PS2, but it was PS1.

And yeah obvious ones FF7, MGS, Wipeout, Tomb Raider, Resident Evil, Silent Hill. 

Also the original Driver, great game.

Emulation will make everything so much easier.

The best game ever made is a PS1 game, so obviously that:

- The Legend of Dragoon

It's available on the PSN classics if you wish.

Some also incredible games with no remakes:

- Diablo (PC game first, couch coop on PS1)
- Digimon World 3
- Final Fantasy IX
- Yu-gi-oh! Forbidden Memories
- Sheep Raider
- Hercules
- Gran Turismo 1, 2
- Ace Combat 2, 3
- Tekken 3

Some other amazing games but those have remakes, modern versions or collection rereleases:

- Resident Evil 2, 3
- Crash Bandicoot 1, 2, 3
- Spyro 1, 2, 3
- Final Fantasy VII
- Metal Gear Solid
- Marvel vs Capcom
- Street Fighter Alpha 3
- Tony Hawk Pro Skater 2
- Castlevania Symphony of the Night

I think that should be enough

The original PlayStation has tons of genuine classics on offer, but Vagrant Story and Final Fantasy 7 and 9 specifically represent my idea of the top-tier, absolute must-play titles.

My honest advice is to skip PS1 and go straight to PS2

Most of early 3D games did not age well at all. Getting a expensive piece of hardware and original discs only to play few games seems a huge waste of money. Look for collections and re releases on PS5, many publishers issues ports that are much less expensive than buying the original discs