Hampshire- Have you played any of the Pokemon games? Beyond the popularity of the anime, cards, etc., they really are great RPGs. The games are more about the mechanics of catching and raising the Pokemon than the story, but it isn’t like the story sucks.
Of the 3 the OP names, i’d definitely recommend Chrono Trigger as well. The original is one of the greatest RPGs of the 16-bit era, and the DS version is pretty much a straight port with a couple extra widgets thrown in.
If the OP is interested in recommendations, i’d like to throw out the Ace Attorney series of games. They’re not so much “RPGs” as dialogue-heavy whodunits, but the plot and characterization is sublime. It’s sort of an anime-ish “Law & Order”.
I recommend the Etrian Odyssey series. It’s along the lines of Wizardry. The first two games are out now, and the third one should be out in the U.S. soon, hopefully.