Picking between PS1 and PS2 will ultimately come down to whatever it costs you to buy either, while the PS2 has some filters to make PS1 look smoother and load a bit faster, it won't make too much of a difference, in my opinion. PS1 Compatibility with PS2 is almost flawless, with the exception of some games. Looking over this compatibility list it looks like you'd be OK with playing them on a PS2.
On either console, you'd need a PS1 memory cards to save game progress.
Connecting to a PC monitor is a separate issue. While converting component signal to VGA is possible through adapters (with varying results), you'd have to keep in mind only the PS2 supports component output, and you may encounter some issues when trying to play PS1 games through it. You can try using converters from composite/s-video to VGA but this will look even worse on a PC monitor.
That leaves you with two other options. The cheaper one would be to emulate the game on a PC. You won't need a too powerful PC to emulate PS1, and the only hardware you might want to get is a DualShock-like PC controller or a PS1/PS2 DualShock and a USB adapter. I believe this will get you the best looking results.
A more expensive one is getting a PS3. It might be an overkill but if you don't have a PC capable of emulating it might be cheaper than upgrading to a new PC. All PS3 models work with most PS1 games (although WipEout 3 does seem to have issues), and they upscale them to run in resolutions suitable for monitors, but again you'll need a component to VGA adapter.