Play whatever Yakuza you can get your hands on. If Yakuza 5 is your only option, play that and play as much as you need to, because Yakuza Kiwami is coming in Summer 2017. As with all Yakuza games, play as much or as little of them as you want. Doing just the story in 5 will be about 45 hours, doing everything might be 100+. If you don't have time for a long game, Yakuza 4 is about 20-25 hours for the story alone. Yakuza 4 is my favourite followed closely by Yakuza 1.
As mentioned, either wait for Kiwami or play Y4 (digital on PSN, on sale often, ex-PS+) before Y5. You can always come back to Y4 and Y5 after Kiwami in prep for Y6.
Thanks for the advice!
I haven't gotten to play quite as much as I'd have liked so far (I have a day job, and just do occasional reviews), but I am totally in love with it. I've always wanted to try the series, but just never got around to it before this.