I'd avoid Dissidia and Crisis Core if you're just dipping your toe into the water. Both are games built for fans, and while you maybe able to follow the stories of each to a degree, it's just not as effective if you don't already have an attachment to those characters.
FF Tactics : War of the Lions is just as Asmodai said, a really great game, but it is a SRPG and is different from any of the numbered entries of the series.
Now I loved FFXII, even though there were plenty out there who didn't. Loved the combat, loved the story, the characters and most of all the world of Ivalice. It's like Star Wars with a faux-Shakesperian script. I would personally recommend it, though probably wouldn't be my choice for a first entry into the series.
FFVII and FFVIII were both excellent games, also highly recommended, but can also look and feel rather dated. If you can look past that, then I would go with either.
I was thinking about it myself, and I'd be hard pressed to disagree with Asmo, FFX would be the best choice. Still feels modern and plays just as well. And only get X-2 if you really
really feel like you need to.

On another note, if you're looking for RPG's outside of the FF series, I'd go with Persona 3 or Persona 4. Very accessible, a lot to do, takes a while to complete and has fantastic stories/characters/soundtracks