If your sticking with whats available on PSN:
RPGs (PS1's #1 genre imo)
Final Fantasy VII, VIII, and IX.
(wouldnt recommend the PS1 versions of FF: Origins, FFIV, V, VI, or Chrono Trigger because the load times are really bad, but all those games are awesome too.)
Breath of Fire IV
Vagrant Story (easily the best looking PS1 game ever made)
Xenogears
Chrono Cross
Suikoden
The Legend of Dragoon
Parasite Eve
Blood Omen: Legacy of Kain
Theres probably a few I'm forgetting
Mega Man X4 and Castlevania: Symphony of the Night are hands down the best 2D platformers from the 90s.
Crash Bandicoot and Spyro are also PS1 staples you should experience.
For 3D action games:
Metal Gear Solid
Syphon Filter
Resident Evil 2
Mega Man Legends
Twisted Metal 2
Medievil
Soul Reaver
Fighting Games
Tekken 3
Bloody Roar 2
Street Fighter Alpha 2
Im not super into sports games, but Cool Boarders and Gran Turismo are stand out titles.
If you wanna know more about any of them just lmk. I grew up playing PS1 and know quite a few of these games inside and out.