@TriforceBunIt's pretty much set up exactly like Lufia II. The overworld is random battles but in dungeons, you can actually see your enemies. You don't have any tools to use and so far there's no puzzle solving. You can still hit enemies by swinging your sword and stun them. Holding down A will also charge up your sword and you can shoot a beam with it which will knock back and stun enemies.
Overall the dungeons are extremely simplified. Every dungeon has shrubs which you cut down. They can either hide traps, treasures or healing points. Walls can also be destroyed by hitting spots on the wall that are "different." Different being they can be covered with vines, torches, etc. Every dungeon is also randomly generated so even if you leave and return, it'll be different again. It's really the downside of the game so far. I have no idea why they went with this design. It's basically the Ancient Cave from Lufia II minus the fun
It's nice to have optional dungeons designed that way but the entire game? Not really.
Still, I'm enjoying it so I guess that's what matters. Part of it may be because this is Lufia after all and I'm just happy to play a game in a series I absolutely love after so long.