As a Final Fantasy lover i will chime in.
It really comes down to what you are prepared to play. The older 2D FF games are decent. FF6 Story is very good. But i find that when i talk to someone much younger than me (im assuming you are younger then me) they just cant seem to give a game like it the time it needs and deserves. Its just an instant 'graphics are terrible' and bin it.
FF7 was the first leap to 3D. I must say i hugely disagree with Libertine. You HAVE to play the original first. But that's because i have the nostalgia of playing the original when it first released. So i can play the latest FF7 games with a whole different lens and appreciation for what they have done with it. Perhaps its different if you have never played the original. Even so to play the FF series and not play FF7 original would just be a real shame.
FF8 and FF9 are both brilliant
But overall i would have to agree with Legend in Blue and go with FFX as a starting point. Easy game mechanics, brilliant story with no overly complicated levelling system with the Spheregrid. Also Blitzball is great.
After FFX i would go into 7, 8, 9 and 12. Then if you have a real love for FF games then go for the older ones.
FF13 i havent played in a very long time but all i can remember about it was how linear it was, something i didnt really enjoy. FF15 and FF16 are both decent but i still find the older games better.
7, 8, 9 and 10 are the pinnacle for me