Exploration in this game is fairly simple.
There are two means of obtaining new dungeons: getting far enough in one dungeon to earn the permit for the next area (usually after the last Dark Angel Statue), or continuing the main storyline.
Aside from the areas you can explore immediately when starting the game, Unlocking the rest of the town areas require advancing the storyline. Remember: Criminals tend to reside in the more shady parts of town, so be careful around strangers.
Advancing within the dungeon often means finishing a Dark Angel Statue's "riddle", which then would remove the fog that blocks an adventurer's path forward. These riddle requirements vary dramatically, but an intuitive player may discover what they must do prior to even reaching the statue.
Later dungeons have multiple floors to explore. A player may use Guide Posts to travel between floors quickly. Each new floor becomes progressively harder. Any Guardian Statues that may have been activated on a floor will remain activated, even if you change floors. However, leaving the dungeon will deactivate all statues.