The games I have and liked best on PS1 are:

Test Drive 5 - My favorite arcade racer. I recently dug it out of the cabinet and fired it up again. Brings back some good memories. It has a decent selection of cars and tracks. For PS1 I thought the graphics were pretty good. There is also a cop mode where you are the cop.

Need for Speed Hot Pursuit and High Stakes - My favorite games out of the whole series (plus NFS 2 Special Edition for PC). They both have a good selection of cars and tracks. Even though HS has a mode where you can be the cop, I kind of like HP better.

Gran Turismo 2 - Possibly the best racer on PS1. Many tracks and tons of cars (over 500). You can highly customize each car and there are plenty of races to keep you busy for quite a while.

Dino Crisis 1 and 2 - DC2 is more of an arcade shooter than DC1. I think DC1 is better because the suspense is there. I always liked the DC games better than the Resident Evil games. The dinos present more of a challenge than a slowly moving zombie. DC1 is definately a game where you don't want to aimlessly go around shooting. Ammo can be very scarce. Plus, I like the puzzles in DC1 better. In DC2 they are pretty much non-existant. In DC2 there are stations scattered throughout the game where you can buy extra weapons and ammo with the points you rack up through killing dinos.

Driver 2 - Some people may love it. Some people may hate it. It is that type of game. I loved it back in the day. I think it is the game that started the 3d sandbox style car jacking genre. Before then GTA has always been 2d. Anyway, there are plenty of missions to do. The game spans four different cities. You can free drive in any city anytime you want to. Some of the missions do need polishing as they can be unbalanced sometimes.