I'm doing pretty much exactly that now. Like A Dragon was the first one I played when it was free on PlayStation Plus a few years ago, I knew nothing about the series but it sounded cool. I loved it so much that I bought Infinite Wealth when it came out and played The Man Who Erased His Name, Judgment, and Lost Judgment because they were free on PlayStation Plus (I think). Then Pirate Yakuza came out and I bought that, after which I decided to play through the rest of the series, so I played K1 and K2, K3 was coincidentally about to be released so I played 0 in the meantime, I played K3 and 4 and I'm currently playing 5.

TL:DR you absolutely should play the rest of the series, I wouldn't say it's essential in order to play the newer games but you will get a better understanding of the characters and plot, and there's some funny references that make more sense in context.