Final Fantasy X is tied as my favorite Japanese style RPG along with 9. It's hard to recommend anything else, because I've never played a Japanese style RPG that touches those two.
None of these are as good, but I'd recommend:
-Final Fantasy 8 (give it time as the game's "system" is really weird at first, one of the most different in the series).
-Chrono Trigger (it's available with I think Final Fantasy IV in a Playstation 1 collection). I hated Chrono Cross, which really has nothing to do with Chrono Trigger as far as I could tell. Chrono Trigger was arguably the best SNES RPG (Final Fantasy 6 is the only other one that could really lay claim to that.) It still probably has nice art and music, and is kind of an honorary Final Fantasy, but has some of it's own ideas (combining magic attacks, battles take place on the same screen you're exploring, etc.) And the time travel aspects are neat.
-Final Fantasy X-2-This one is both good and a disapointment. It's incredibly trite compared with the real Final Fantasy games. X had all kinds of things they could have built off of plot wise, but this really doesn't resolve anything from X. It also recycles practically all the graphics and the game world.
Yet at the same time it has a completely different "system" (as different as any of the FF games are from each other) so it does feel like a new game and is rewarding to play. Just expect to be severely disappointed by the story. Well worth $20 though.
-Golden Sun games-I agree that these are pretty great. Not QUITE up to Final Fantasy gameplay standards, but close, and some of the best RPGs this past generation. Their twist is you use the summon-esque creatures outside of battle too to solve puzzles. The battle graphics are actually very flashy and impressive. And the two games are actually connected (for once). You use an (incredibly long) password after beating the first one to import your same party into the second (though you don't actually join up with your party from the first game until really late into the game). If I'm remembering correctly, the second game starts with you playing sort of the side that was the bad guys in the first one.
Golden Sun is by Camelot, and it's their only RPGs since way back with Shining Force. Graphically these even have the same look as Shining Force, except much flashier, better color, resolution, etc.
On PC, there's tons of other RPGs that play really differently and are more open ended than Japanese games. Nethack is free, and is literally one of the best games ever (despite using ASCII characters for graphics-I'm not kidding, it really is one of the very top games ever made). Deus Ex is also brilliant, and although some people call it a "first person shooter" it's not (at least no more than calling Metroid a shooter, which is equally silly). It does have FPS mechanics, but it is an RPG, and the most amazing thing about it is how open and free the gameplay is-how many different ways you have to accomplish the same thing. The developers tried to give you freedom rather than the typical linear rail 99% of games are on.
Oh, and Dragon Force by Sega/Working Designs and Panzer Dragoon Saga are two of my other favorite RPGs ever made...but unfortunatly they're both only on the Saturn, and both super rare, so finding them would be a challengle. They're up there with the best of the Final Fantasy series though. Dragon Force is my favorite strategy RPG ever (played it for hours a day for MONTHS, and it's nothing like the typical Shining Force-esque games). And Panzer Dragoon Saga is just amaing, and nothing else is like it.
