In my opinion Final Fantasy 9 is the perfect place to start. In short, it's the most moderate Final Fantasy experience you could possibly get.
It's pretty much right between the old and the new Final Fantasies. It doesn't have the extremely good graphics of the new ones, and it doesn't have voice acting, but it's not 2d like 1-6 or extremely blocky 3d like 7. This gives you a sense of how the new Final Fantasies will look, while not spoiling you on their modern graphics. FF9 is also a sort of retelling of the rather simplistic story of the original Final Fantasy.
Concerning difficulty, it's not the hardest game in the series, but it's certainly not the easiest either. To me, it just seems like the best starting point because it's not extremely old, it's not new, and it's got that classic Final Fantasy story. I'm not even saying it's the best in the series. From what I've seen, that's generally hotly debated between 6, 7, 10, and Tactics. It seems like 6 is the current flavor of the year, but that used to be 7. I say play 9, and then just work your way through the rest of the series as you see fit. Either starting at the beginning, or jumping right into the most popular ones.
Most people will just tell you to start with their favorite though.