Extremely bare bones playthrough would be just doing Yakuza 0 and then 7. Those would give you an overview of the two main gameplay styles the series has to offer.
If you want to play more then you probably want to finish the brawler games first before moving on to 8 since it has spoilers for the whole series.
The Judgement games are also good so you might want to try the first of those to see if you like it. Story is completely isolated from the Yakuza games though so I'd investigate this at your leisure.