Darius Gaiden is a good 2D spaceship shooter, I have it on Saturn and I imagine the PSX version isn't much different. Nice to look at, fun to play.
Get Front Mission 3, it's one of the best games Square ever made.
Carnage Heart is probably very cheap, so try it, it's a game where you sort of program battle robots and then watch them fight. Some strategy involved too, I think (It's been a long time since I had a PSX)
Destruction Derby 2 is the best driving game I have played. I don't think there's any other game where you can finish first with one tire, steering with opening and closing the doors, just because everybody else is in even worse shape.
Get Bio Freaks if you have friends to play fighting games with, it's ridiculously gory and stupid in many ways. You can maneuver in a 3D-arena much better than any other fighting game I have played, it's fun to fly around trying to shoot your opponent, dodging bullets and trying to get to him. Not a great game so don't pay much for it, just fun to laugh at.
There's probably some good fighting games for the PSX too, Tekken, Street Fighter, Darkstalkers etc, but I wouldn't play them without a proper arcade stick. Come to think of it, neither would I play spaceship shooting games without one.
Have you tried the Strike series, there's at least one (Soviet Strike) on the PSX?
Return Fire, Mass Destruction. Eh. Games mentioned ITT.
Get a demo CD that has Psychon on it, it's a simple game where you shoot some guys.
Most other games I can think of, like Little Big Adventure, Crusader No Remorse etc. are also on PC.