Well, as far as what you start with it doesn't really matter too much. With a DS you can play FF3, FF Tactics (for the GBA, but obviously a those work with a DS), Dawn of Souls,FF4, and then XII: Revenant wings. The Xbox 360 can play FF11 (plus rise of the Zilart Treasures of Aht Urhgan, and Chains of Promathia, wings of the goddess all branching from FF11), FF13 and FF14. Of course that's the 360, and if it's not a 360 then....none.
You can't really play a lot of the main games (being the ones with just a number) but mostly the spinoffs. The most popular ones are for the playstation/playstation 2 (it's also nearly impossible to find a copy of FF7 that won't cost you a decent chunk of change OTL). There's a fair amount for the PSP, and everything before 7 are for the nintendo/super nintendo (although you can get anniversary copies of 1-6 for the PSP)
REMIND ME ON FRIDAY, WHEN IT WON'T REQUIRE ME TYPING, AND I CAN SHARE WITH YOU MY KNOWLEDGE OF FINAL FANTASY AND ITS WONDERS.