Dude, I don't care what anyone else said, the best game to get you instantly addicted and also show you one of the best parts of the genre is Chrono Trigger. It's not terribly difficult with little to no grinding and it's plot is epic and the combat is way more action packed with one of the first active time battle systems that also used your characters location on the screen to effect attacks.
Also, if you're into weird trippy shiz like me, might I recommend Earthbound. It's insane, and beautiful.
Of course if you like Chrono Trigger I'd move on to the Final Fantasies (Start with VII or VI, possibly IX. X if you need semi-modern graphics) or Chrono Cross, or possibly Dragon Quest though that stuff is sometimes painfully old school (maybe start with VIII). There are so many good ones beyond these mainstream series, however these are really the best introductions as the high production values and quality stories make it easier to adjust to JRPG staples like turn-based combat and random encounters which can be somewhat annoying (although I still love the levels of strategy open in the turn based games, with Dragon Quest VIII being in my opinion the best example because it was also rather hard at the higher levels requiring well thought out strategies.)
Oh, one other thought, as it was the first JRPG I played, and I'm sure a lot of people can say the same: Pokemon? The plot is garbage which is why I can't enjoy them anymore but the battle system was epic in it's depth, plus there is nothing better than fighting your friends with your characters. I wish more JRPGs would do this, I don't care if it means Cloud vs Cloud so long as the build is different and the winner still gets bragging rights.
Man, why can't there be a turn based Street Fighter RPG. =<
OH, and I forgot to mention, the best version of Chrono Trigger with a more mature translation and extra dungeons (that aren't that great to be fair) came out on DS a while ago. Let me put it this way: I sold my DS but I didn't sell my copy of this as I know at some point I'll get another DS pretty much just to play it.