Ignore 13, its less of a game and more of an 'interactive movie', and I'd also avoid 12 unless you're *really* into grind-tastic single-player-psudo-MMOs.
Luckily for you, most of the older FF games have been reissued for PS1 and are available in good collections
Final Fantasy Anthology = FF5 & FF6
I'm a HUGE HUGE HUGE fan of the story of FF6, just play it and you'll see its much better than FF7
Final Fantasy Chronicles = FF4 & Chrono Trigger
Though slightly misnamed and its a bit odd they didn't have different ordering, but meh. Chrono Trigger alone is SO worth it, and it'll make you wonder why the hell you EVER put up with random battles in other games. Imagine a world map where you can actually explore a bit and you are not going to run into random battles every 5 seconds.
Final Fantasy Origins = FF1 & FF2
You can't call it Final Fantasy without the first two games. Oddly enough FF3 still hasn't been re-released (to my knowledge) so its only available via a japanese rom and translation patch, via emulation.
I'd say if you aren't sure or need to know if you'll even like the games, then get the roms and emulate them to get a feel for them. I'd say FF6 and Chrono Trigger are well worth it, but the other ones are kinda...meh. They have their fans. Also you deal with load times with CD media whereas the original cartridge versions never had load times, just regular brief pauses between areas, so there's that issue to consider.
EDIT: Final Fantasy 2 wasn't horrible, it was innovative, incredibly innovative and ahead of its time. Granted the battle mechanics got a bit odd from our usual perception of 'leveling' but it actually made sense. Use something and it gets better while something else is worse. In reality though it sorta forced you into class roles anyways but made you really work for it instead of just gaining levels and buying stuff. Its different but by no means horrible.