Well, PS2 can play any PS1 game.

Also, I think there is a super console of sorts for Sega/Nintendo for earlier eras. I haven't ever used one though, so I've no idea how well it works.

Some of the more popular PS1 games have made it to the Playstation Network. They generally work on PS3/4 and the PSP/Vita.

Some games have made it to Steam, and GOG has some as well. I think FFVII, VIII, and IX are on there. Possibly X and 12. I haven't checked.

--edit: For PS1 games on the PS2, you'll need PS1 memory cards. There should be third party cards you can buy. I bought those back in the day since they were larger and cheaper than the official Sony ones.