I'd say 3 - no one actually remembers one or two (two being a duology) because one got a really shit localization and two was the Deadly Premonition of RPGs (weird abstract shit.. just watch the trailer for the new PSP release).
3's probably the best place to start because there's no demons, some idiot in a mask, any philosophy and most certainly the most intentionaly dark plot of them all (Persona 1 steals the darkest plot because of it basically turning into Battle Royale halfway through and no-one noticing).
Also, it's hard to be confused by a: Sunday, kill monsters, Monday, take girlfriend to restaurant, so on so forth.
Or you could put your hands in the air, start with Shin Megami Tensei (the first) and have no fucking idea what's going on.
HISTORY LESSON TIEM!
Megami Tensei were books that were the foundation for the series. They were full of rape, murder, more consensual sex, and of course, demons (yah....)
Majin Tensei had two games, both were dark and employed the same tone as the books - they sucked.
Shin Megami Tensei had:
SMT: Some kid deciding the fate of the world.
SMT2: Some adult deciding the fate of the ruined-again world.
SMT3: Some demon deciding the fate of the WHY DOES THIS KEEP HAPPENING?
SMT3.5, Strange Journey: Some guy from -insert nationality here- with a team of racial-stereotypes going to investigate a hole that is literally eating the world inside-out. Also, demons at some point.
SMT 4: Is on the way.
Persona 1: Turns into Battle Royale halfway through, all-child cast doesn't seem to notice.
Persona 2: Innocent Sin wasn't released in English until November 20th last year. It was too late, ofcourse - it wasn't released because of the ability to start a lover's chain with anyone in the party except from Yukino, who leaves about 3/5 through the game.
Persona 2: Eternal Punishment is a confusing mess without Innocent Sin, and as such, just to be douchebags, released it in English but not Innocent Sin.
Persona 3: Was a turning point in the series - adopted day-to-day simulation.
Persona 4: Expanded on this, and added jobs and added two new Personality skills, Understanding and Dilligence and renamed two.
Persona 5: Is coming.
Persona 4 Golden: Likes Persona 3 FES and Portable, is a remake type-thing, but instead of an epilouge, adds a gardening ability, improved fishing, introduced a couple jobs, some events, two social links and a bug-catching minigame too.
Persona Arena: The epilouge to Persona 4.
Now, the more obscure side:
Lost Bible: As its name suggests, it's more about Christianity and Catholic mythology than anything else. It was really popular in Japan for a little bit of time, and was for the Gameboy Colour.
Tokyo Apocalypse: It had the Valkyrie Chronicles problem in that enemies not engaged in combat could still attack you, but not the other way around. Also, for shock-effect on the artists' part, uncensored boobs. Literally the only reason it's remembered.
The Devil Summoner series: Consists of four games, one and two, and The Souless Army and King Abbadon. 3+4 are the best-received and most-remembered.
Soul-Hackers: Not so obscure back in the day, was really good and is being remade for 3DS.
Some shitty game on that Nintendo thing that blinded people - Virtual Boy.
So, the starting out point..
If you like a psychological delving and like the idea of trying to keep a sense of social normality when you keep getting attacked by monsters at midnight, try Persona 3 and 4.
If you want to know where the dungeon-crawler and and social-simulation fused together in a weird way, try Persona 1+2.
If you want a game almost entirely focused on being Intelligent enough to be diplomatic -or-, take a machine gun and rattle the enemy with bullets until it dies, then try the main series.
If you want a mash-up, try the Devil Survivor series. It had around 7 endings, tons of playable characters, a demon-fusion system making a comeback in the form of jamming your demons into a DS and taking out the thing that comes out, and a strategy/turn-based JRPG hybrid, then try this series.
If you want a real-time, action game with RPG elements and strategy try the Devil Summoner/Raidou Kuzunoha series.
I'm sure you'll find something you'll like.