I'm going to be boring repetitive and predictable here. But there's a reason some of these games come up in every thread.
-Pokémon, Nintendo DS, if you're not a pokemon person; start with black/white. There's a reason it's this successful after 15 years. One of the deepest competitive rabbit holes out there. The level of complexity that goes into the high-end battling combined with the simple addictive progression make it one of those series' that truly is suitable for all ages. Here's [http://www.smogon.com/forums/showthread.php?t=3482311] an example of the level of detail we're talking in a 'rate my team thread' it will go waaaay over your head but serves as a reasonable example. Full honesty here, I've spent more hours in pokemon than anything else and I played WoW for years. I like the series that much. In fact... I think I'll go play some now.
-Chrono Trigger, Nintendo DS, fan favourite, classic, some say best SNES JRPG- I would be inclined to agree with that. Fun characters and a pretty cool if clichéd story along with solid ATB fights and NO RANDOM BATTLES make this one to play. Then comes the art, much of it done by Akira Toriyama (if you don't like his style this may put you off, the character designs are very much DBZ) and it it beautiful. I still regularly listen to the soundtrack and various remixes to this day. Play this game.
-Star Ocean (Till the end of time), PS2, one of the last games released by Enix back in the PS2 days, and it's one of the best in my opinion. Think final fantasy with a real time battle system and some slight tweaks to the formula. But honestly it's pure JRPG at heart. Another case of no random battles being a selling point for me. Vastly unappreciated.
-The World Ends With You (TWEWY), Nintendo DS, This game can't get enough praise in my eyes. A fresh and interesting take on the genre. Unique battle system very dialogue heavy story with no real player agency, but some excellent characters and a gripping story get you hooked. It's a very 'marmite' game, you either love it or hate it. It's nowhere near as loved as some of the others in my list but it's one of the first ports of call for me as far as great JRPGs go. Best thing SQUEENIX has made in a decade.
Then you have things like the Tales series, Xenogears, Secret of Mana, Persona series, Earthbound, Disagea (however you spell it) and many more. Frankly JRPGS are far more interesting outside of final fantasy in my opinion. Modern FF is almost a gateway drug to the great experiences out there. The Escapist has great and varied taste in the genre so I'll let those who know more about these games say it.
Oh, and if you're up for a Strategy RPG, you literally can't do better than Fire Emblem (series). Awakening is the current incarnation and by far the best. It's got the highest metacritic rank of any original (IE: not remake) 3DS game and is just amazing.