There's a good chance you've played a persona game before, but there's a much smaller chance that you've played one of these. 

These games improve a lot on what Persona 1 started, particularly by making the characters and dungeons more interesting. Persona 1's first-person dungeons are certainly interesting, but a lot of them feel the same as each other, and I for one don't really remember most of them. By contrast, off the top of my head I can remember the sewer dungeon, the music store dungeon, the mine dungeon, the dungeon where you control a boat, the dungeon that takes place in an alien spaceship, the dungeon that takes place in a research facility, and the dungeon that takes place on a cruise ship, and I could probably name at least a few more if I thought about it for a bit longer.

Of course, there is the question of whether you should play the PS1 or PSP versions of these games. If you don't want to use English patches, you'll have to play the PSP version of Eternal Punishment and the PS1 version of Innocent Sin. However, if you are willing to patch the game to play it in English, I'd recommend playing the PSP version of Eternal Punishment for no other reason than that I think both of these game's remixed soundtracks are better than the original. It should be noted that this isn't like Persona 1, where the North American PS1 release has missing and unecessarily changed content, so if you find that you like the PS1 soundtrack better, you won't really be missing out on much.

Since this almost tripped me up when I first wanted to play these games, I thought I would also make clear that Innocent Sin comes first in the duology, and                                                          should be played first. Don't play Eternal Punishment first.