Chrono Trigger and Persona 3 and 4 have already been mentioned but it's well-worth recommending all three another time. I'd specifically recommend the DS version of Chrono Trigger, which has the added cutscenes from the PlayStation version as well as extra content, mainly in the form of extra endings. And while getting a hold of a PSP version of a game might not seem that great at first, the PSP-exclusive Persona 3 Portable is easily the greatest version of that game.
Aside from that, I'd recommend:
Dragon Quest VIII: Journey of the Cursed King - this is my favourite RPG of all time. It has a massive world, gorgeous anime graphics with characters and monsters designed by Akira Toriyama, a fantastic story, one of the best soundtracks and absolutely perfect classic turn-based combat.
Breath of Death VII and Cthulhu Saves the World - you can buy both of these together for a tiny amount of cash on Steam and they're brilliant. Old-school pixel graphics, solid turn-based combat and a cracking self-aware sense of humour.
If you have a 3DS, Bravely Default is a very good purchase. It's basically a Final Fantasy game in all but name but no less worth a play, if only for one of the best attempts at a job/class system since Final Fantasy V.
Finally, you could give the GBA/DS Golden Sun games a shot. While I'm not a fan of the soundtrack, basically everything else about it works especially well, particularly how the class and magic systems change by equipping the collectible Djinni.