Chrono trigger and Final fantasy V (us)... my pretty girl...

EARTHBOUND! Great for girls and boys alike

Earthbound, Chrono Trigger and Final Fantasy VI. (III) (Also consider Final Fantasy IV-II) After that, if you want more, try out Breath of Fire II, Lufia; Rise of the Sinistrals or Secret of Mana. Those are a little more generic, but they're not classics for nothing. But seriously, go with Earthbound first, I'm sure you'll appreciate it. (It's going to kick your face's ass if you're not used to RPG's though.)

Chrono Trigger is great. The regular encounters are far too easy, but the boss battles more than make up for it. It also has multiple endings, complete with a New Game + if you can finish with the required criteria. Lufia II is lots of fun, but like CT, its regular encounters are really easy. (Though perhaps I overleveled, though if I did, they should have been a bit less generous with experience points.) The boss battles aren't quite as great as Chrono Trigger, but the boss battle music is excellent, so that's a small consolation. Terranigma is a great game as well, though it's an action-RPG. The graphics are quite nice and Ark, the main character, has a number of moves that he can perform. The ending is also very touching. Definitely something to check out if you can. And then, there's the obvious; Final Fantasy IV-VI. If you can get into it, Paladin's Quest can be fairly interesting. I've yet to finish it, but after giving it my third or so chance, I think it's interesting enough to warrant a play-through. I don't particularly like the Ultima's on SNES (Though Quest of the Avatar is excellent, that is if you like "Western RPGs") and they play more like a real-time action dungeon crawler than an RPG, but it might interest you, nonetheless. Romancing SaGa 1 - 3 are also fairly interesting if you can find a translation. I've yet to beat any of them. They are extremely difficult. (Though not as difficult as later SaGa games, such as SaGa Frontier II or Unlimited Saga (which is my personal favorite, despite being critically panned.)) Though, if you have the means, the first Romancing SaGa was remade into a PS2 game, and has a revamped battle system that plays much closer to games later in the series (which, in my opinion, is a good thing.) I'm not the biggest fan of the Secret of Mana series, but you might enjoy it. It includes (on the SNES only, later games were made on other systems.) Secret of Mana, Seiken Densetsu 3 (Or what would have been known as Secret of Mana 2 if it were released in America.) and Secret of Evermore, though that's really a branch off of the games rather than a part of the series. It's an action-RPG with a sometimes frustrating (at least for me) stamina bar system; After attacking once, a bar drains down to empty and fills itself back up. Attacking while the bar is not full results in a weakened (or sometimes severely weakened, depending on how far the bar has been filled) attack. I personally think the GBA remake (Sword of Mana) of the Gameboy game that started the series (Final Fantasy Adventure) is much better in terms of gameplay, but there are many that strongly disagree with me.