Well,in my humble opinion I recommend you to play the most significant ones,you say that you have a PS3 with full retro-compatibility ,then look for PSone compilations (they have the original Super Nes and Nes versions,plus some fmv and stuff):
* Final Fantasy Anthology ( Final Fantasy V & VI )
* Final Fantasy Chronicles ( Final Fantasy IV & Chrono Trigger )
* Final Fantasy Origins ( Final Fantasy I & II )
Then you can play FF7,FF8 and FF9,the three of them are very good and they're for the Psone.
Then again you should check FF10 & FF12 for PS2.
If you want skip to play them all,then you must play the essentials and most significant ones,that for some reasons they're considered as the franchise's pinnacles,then you have to take a ride with:
-Final Fantasy 1: The one that start it all.
-Final Fantasy 4: The first on Super nes,with a more development history and gameplay,memorable history and great characters.
-Final Fantasy 6: For many (including myself) the best RPG of the history (and obvious from the whole series) with an incredible plot,unforgettable characters,awesome gameplay and some of the most beautiful sprite graphics on a game.Just perfect.
-Final Fantasy 7:The first one for the PSone and because of that,the first jump from sprites to 3D graphics,the one that brings RPGs to the mainstream,epic plot,the most popular FF crew (just who the hell don't know Cloud or Sephiroth), the first FMV introduction to the series,well,you have to experience for yourself.
-Final Fantasy 10:The first for PS2 and getting an awesome history and better graphics according its time,you can't skip this one.
-Final Fantasy 12:The last installment on FF history and perhaps the most controversial one,why?because the gameplay mechanics,a big evolution (for some people a wrong turn) from past games.Is very worthwhile and you can get used easily to graphics and sound being the most recent release (2006).
I'll recommend to check all of them if you have the opportunity,FF games are a wonderful and unique gaming experience and should be played by every RPG and gaming lover.Almost any of them could be played on your PS3,just buy them from SE official shop or go to retailers that have used games,if you have a PSP or a DS you can play the remakes of FF1,FF2,FF3,FF4 or more complete FF5 & FF6 versions but you can play the original versions,get for sure that these games are going to be one of the most grateful gaming experiences for you.
Well,if you like RPGs of course
and if you don't,after play them you could start to like them