It all depends on your taste. Each Final Fantasy game has it's own separate story, so it doesn't matter at all where you begin.
On the Nintendo DS, you can play Final Fantasy III or IV, which are both good. Final Fantasy III is strictly turn based, while Final Fantasy IV has an Active Time system where you can attack at any point (without waiting for your turn) and waiting for a bar to fill up before you can attack again. Final Fantasy IV has more of a story to it, the third one doesn't really have a moving plot like the fourth.
On the PS2, I would suggest Final Fantasy X. This one also features the Active Time System, I believe. It got many good reviews, and where your finished there is a sequel to it. I heard it has a good story and it also features a mini game called "Blitzball", which is like the soccer of that particular fantasy world. But its played in a sphere of water.
On the PS3, start with Final Fantasy XIII. I think this uses Active Time also, but I'm not sure. Also had good ratings, and you should get this one if you're looking for a next-gen Final Fantasy. This one has a sequel that will be coming out soon.
If you're also interested in the spin-offs, then check the Wikipedia page. I only know about the numbered series.