BACKSTREET BILLIARDS / ASCII / PlayStation 1
Backstreet Billiards is a solid but pretty standard game of pool paired with a mild "visual novel" element in the single-player story mode.
By modern standards, the game is nothing remarkable, but when it came out in 1998 it was the best billiards option for home consoles with the widest range of options. The physics are very good, the aesthetic presentation is very basic but the fundamentals (like ball sounds) are solid. All the major rule variants are present, as well as a carom table option with five of the most popular rule sets for that particular billiards branch.
If you're not all that familiar with billiards, it's easy to jump into. The basic mechanics of the game are very intuitive with a "ghost" cue ball path that indicates where your shot will go, and a sliding power bar reminiscent of golf games being the only thing you need to do to activate your shot. You can do a couple of advanced things like manually moving the strike point of the cue to anywhere on the ball surface, but that isn't necessary to play, or even to get through the story mode. The one big "training wheels" thing it's lacking is pathing for the balls you hit; you only see where the cue ball is going to go, nothing after it makes contact with other balls.
Aside from just being a comprehensive and competent sim, the big hook here is the story mode. It's basically in the format of a visual novel, but it's very threadbare and simple. You play as some unnamed kid whose father was the greatest billiards player in all the land. His cue was stolen from your house, and you are rummaging through some city looking for it. Of course, you have to beat people at billiards to get information out of them.
The whole process is totally linear, you just jump between several locations in an unnamed city (that I thought had a weird Parasite Eve vibe, but maybe thats just me) playing people in various game types. As a new player with no particular experience you can just jump right into this, as the computer AI is surprisingly bad all the way up to the final challenger, who sinks nearly every shot and who you'll probably have to beat by going first and going on a perfect run in at least a couple of games. The story isn't at all interesting, nor is the dialogue or characters, it's all about as basic as it gets. It's a little something extra for a solo player to do, however, and unlocks more computer opponents for the other game modes.
This is a pretty good choice if you just want a smooth and easy-to-play pool option that covers just about all of the major competitive rule sets. There are certainly more robust options out there for other platforms, but this might be the best overall choice specifically for the PS1. If you just want some basic and fast pool that gets you into the action quickly, this game is still super-cheap at the various online stores as I write this, even for a new copy.