I have been interested in this series for a while now and I was wondering if the older games hold up or if I should start with the newer ones, and if so how lost would I be with the story?    

You can start with 3 and be fine. They throw around lots of names and stuff but if you watch the 'Reminisce" stuff you should be fine.

I remember liking the first one, so that's a fine starting point.  Like a lot of stories that come out of Japan, it requires a lot of patience. If you like JRPGs, you might like it. The fighting is always fun, and at least in the first game, the payoff was awesome at the end.
 
I've been thinking of playing the second or third as of late as well.
 
Having said that, Amazon has Yakuza 3 for dirt cheap! I just bought it!

Yeah, I recently started with three, and it has has lengthy recaps of the stories from the first two games, and it did a superb job of changing me from someone who knew nothing about the Yakuza series into someone who knew quite a bit about it.  
  
Just remember, it can be a really slow paced game at times. Almost maddeningly so. But on the other hand, the fights in the game are awesomely brutal, and really fun. 

You should start with 4.  They have that Reminisce feature that will get you up to speed. 
 
Though if you want MY advice, DON'T get into Yakuza.