As Nitrus and others said, stories are not related apart from some recurring (mysterious) characters and their "home", but they're enhanced in comparison to previous sequels. For example, Persona 3 introduced relationships, but Persona 4 created romance options; P3 had one dungeon; P4 has several different dungeons... In fact, if my memory serves me right, in P4 you'll visit with your friends the school from the P3 story. And that's also the reason for SMT and Persona's popularity. If you play RPGs, you know that 90% have fantasy/medieval settings, 9,9% are science fiction and 0,1% are contemporary settings.
And when Persona 4 became such a huge hit that surpassed the Shin Megami Tensei franchise in popularity, they dropped SMT in all future Persona games titles. Now, in P4 is really hard to get the true ending without a walkthrough on your first playthrough - so, you'll either have 2 or 3 playthroughs (which I would recommend) or you should use a guide,
You'll probably notice Persona 3 has slightly worse reviews than P4 or P5 but that's mostly because of old Persona fans - Persona 4 had 2 versions - PS2 and an upgraded PS Vita version (that's the one remade for PC), but Persona 3 has several versions - PS2and updated PS2, visually the best, PSP... The thing is, while the PSP version is story-wise the most complete, instead of movie cutscenes, you have static visual-novel-style screens for cutscenes. So, the game is just as good as other Persona games, but we would all prefer if Sega and Atlus invested more time and energy and gave us a combo of the best thing from those old versions.
P5 didn't play yet, but I know I won't play Strikers because that game basically ignores your relationships from the main game and I don't like such things.
Apart from SMT Persona games, I'd definitely recommend you SMT Nocturne - there is a reason why that is the only SMT game (apart from Persona's series) remade for PC.
For the end of this long post, something for P4 fans: