I just got into the series a few weeks ago. I'm amazed I never even tried to look into the series before since I'm a fan of a bunch of other Sega games. Funny thing though, I had a couple of the PSP spin-off songs on my computer, but that's it. Maybe I wasn't interested before because I saw people mostly focus on the funny sub-stories, but after playing myself, the story is intense. Especially since 1 and 2 were written by someone who's never done work in video games, but has written for numerous yakuza themed movies, the influence definitely carried on to the later games.
I'm currently playing 0, I think it's pretty clever how they made Kiryu and Majima's progression sequences play out almost exactly, except everything else is different. Majima's really fun to play as since his fighting styles feel more natural. Also, the cabaret club minigame is addicting and I'm glad it brings me in a lot of money.
I'm really excited to start playing Kiwami, especially with the updated engine, PS2 Kiryu looked like a mix of a chimpanzee and John Travolta at times.
Also, I recently played the Kiwami 2 and 6 demos and I'm really loving the new Dragon engine.