The best Persona game is Shin Megami Tensei if... and I'm not kidding. But if I were to kid I'd say it's P3FES because the story is beautiful and because I actually like the battle commands system.

re: pedanticbiomasochist
Sadly Tartarus is a bit of a love it or hate it thing (though I'm not sure if there are many who love it) but I think it could be more tolerable if you were to play Reload because of how easier the game is. Not everyone has the patience for it and that's fine. As for the other games, I'm sad to tell you that P4 isn't much better in that regard. The dungeons aren't as lengthy as Tartarus can be but they also have the same repetitive feel gameplay wise, and the times it tries to shake things up are disappointing to say the least. On the other hand, Persona 5's dungeons rock since they're all uniquely designed and have actual puzzles in them.

When it comes to social links, I think all of the games are kind of on the same level, in the sense that there's going to be some you dislike and some you like and that's fine. Personally I'm not a big fan of P4's social links but I know some people love many of those so it really depends on the player. I really like P3's social links and some of them do get a lot better towards the end of each storyline, so I think it might be worth pushing forward. The game also has a stellar ending unlike anything I've seen in other games so I think it's worth sticking until the end for that too.