"There are three 'modern' Persona games" is the popular understanding - 3, 4, and 5. I have played all of them (in particular, the enhanced versions of each). A brief rundown of my thoughts:
Persona 3 FES: Does a good job of establishing an ominous feeling of foreboding in the background. Strong intro at the start as a result and the conclusion is the most compelling of the three games, but the second act suffers from meandering and from weak, secondary antagonists. Mechanically the game is the roughest (unsurprisingly) but the restrictions placed on you in combat can make for some fun encounters from time to time.
Persona 4 Golden: Nails the cozy, small town feel perfectly. The scooby gang you assemble to investigate the mystery of the town has great chemistry too and, together with the first point, is the main reason people rightly love this game so much - It's the best 'what if', fantasy high school simulator game.
Persona 5 Royal: It's just Persona 4 again but worse. It's the same scenario at its core but plot developments tend to be dumber, and the cast doesn't have the same chemistry as the last game's to make it all come together (it's tough to summarize without going into a diatribe, but I think the characters, while well-executed, are at odds with the design philosophy that had been ripped from Persona 4). What Persona 5 does have going for it though is that it is ridiculously stylish. So, so stylish. Every visual and mechanical element has been improved from the last entry.
All that said, I think 4 might be the best place to start, as it's the best one imo. Allot of people prefer 5 though (and have said as such here). If you're still interested after playing one of the games, then ask for more and recommendations can be made as to where to go from there (I want to believe the core of Persona 3 stands the test of time).