I was about to say that Kiwami would be the place to start, since it's a remake of the first, but Harry makes a good point on starting with Zero. I'm about halfway through the story of Zero now, and it occurs to me that playing Kiwami after this is really going to make everything between Kazuma and Nishiki resonate a helluva lot more than it ever did in the original game, really make it "pop" in ways it never had the time to in the original PS2 version.
Only drawback is that, from what I hear, the sidequests and whatnot are pretty much all exactly the same in Kiwami as they were in the original Yakuza 1. That is to say... they aren't going to be quite as fun and charming as the ones in Zero.
But all the same, for main story purposes, Zero's definitely the best starting point.
And then, as luck would have it, Kiwami 2 releases in August.
Zero, then Kiwami, then Kiwami 2; it really is a pretty perfect time for people to jump on board with the series now. I've been following it since the beginning, so it's not the kind of thing I have to think about much, but just like what a lot of people are saying about Shenmue HD, I'm almost jealous of anyone getting to start the Yakuza series right now. The devs have the series down to a science by this point, and it's all pretty great (long as you don't burn yourself out, like I did a while back; only just recently got the spark back).