All the numbered personas are independent from each other. There are some games that are basically fan-servicey crossovers (Persona 4 Arena, Persona Q), so I recommend starting with either 3(Portable), 4(Golden) or 5(Royal). You can pick just one and go!

Depending on how you're feeling:

• Persona 3 has an amazing moody vibe to it: I'm sadly afraid it'll be a little toned down in the remake in favor of something more akin to P5. It can be grindy, but you can play it on easy if you can't be bothered. Maybe it's because it's the first one I played, but I still deem it my absolute favorite of the tree.
• Persona 4 is something of an in-between, but a banger nonetheless. It's a game in which the usually upbeat soundtrack is placed in direct contrast to a mysterious foggy countryside town.
• Persona 5 is, in terms of gameplay and fluidity, the absolute best. I highly suggest you NOT to start here simply because after you get used to the gameplay of P5, dealing with P3/P4 will be slightly harder imo; those are still highly relevant to play nonetheless: they all have very different stories, very different characters, some may be for you while others may be not. For me, P5's story tried to be so grand that it failed in delivering the same "personal connection" that the player could feel with P3/P4. Or maybe I grew out of highschool settings by the time P5 came around, who knows.