I just started playing the Japanese version on the PS1. The graphics have aged pretty badly on that one. It's a mixture of polygons and sprites and you get the worst of both worlds. All the graphical enhancements look nice on the 3DS.
You basically unlock new islands (which are their own self contained worlds), do a "save the world" quest, and then the island you saved gets added onto the main map. It's heavily implied that the worlds you travel to are in the past, and the main world is the present. So it's like Zelda: Ocarina of Time, Dragon Quest 6, Zelda: Link to the Past, Chrono Trigger, etc. where there's 2 worlds and the actions in one affect the other.
So far there really isn't much of a story. All the islands you travel to have so far had their own self contained stories so once you finish one you go to another one. It does feel like it's starting to get repetitive. I mean how many mini-save the world quests can you do? There isn't much of an overarching story yet either. But the rest of the game does have the usual DQ polish and focus on exploration and lack of handholding that most JRPGs don't have.
The one thing I never got about the DQ games is the music. The typical Koichi Sugiyama music is always just bland background filler and never seems to rise above anything more. Nobuo Uematsu, Yuzo Koshiro, among others have made music for decades that is entertaining to listen to as well as adds something to the game but there has never been a single memorable Sugiyama tune from a DQ game that I can think of.
One thing I do like about this one is that they made it a bit more puzzle focused than previous games. It takes away a bit from the mindless grinding and battles. The fact that you don't get into any battles for the first hour (though it took me almost 3 hrs to get to a battle) is a good thing.
I really want to play the 3DS version after seeing some shots but I'm too far into the PS1 version to start over.