The Persona games, while all existing in the same universe and sharing many play mechanics are largely standalone. You can pick up and play any of them and not miss anything storywise by not playing the others. That said each subsequent game adds certain QoL features which, once gotten used to you’d be hard-pressed to play without. I’ve played 3, 4, and 5, in that order and I would recommend the same to anyone looking to get into the series.

The Yakuza series on the other hand all follow a set timeline, and though each game has its own plot they all exist in an evolving overall world. They should be played in order, at least 1-6. Zero is a prequel though it was made around the same time as 5. As such it has several references to future events which are not really spoilery, but without playing the other games you won’t necessarily get them. But they are not vital to the plot and the game can be enjoyed first. That’s what I did and never felt like I was missing out on anything. 7 is a new start for the series and while taking place in the same world and featuring guest stars from other games it is not closely tied to the other games in the series. Personally, 7 is my favorite of the bunch with Zero a close second.