There is no world map. For plot relevant things, like dungeons and towns, you are usually given rough directions or sometimes even brought there directly. However, that is not always the case. Sometimes you need to find your own way and explore the world.
As for how to find the first dungeon: once you exit the city, go west and cross the bridge. There should be a farmer around here, who can give you directions, if you still need them. From there, go in north, north-eastern direction. You should eventually come across a lake. If you walk north along the eastern side of the lake, you should find an old bridge. On the other side you'll find the dungeon. While crossing the bridge, there is a chance that it'll break and hurt your party leader. You can also simply avoid the bridge, and simply go around the mountain in the north. You'll eventually arrive at the dungeon from the west. However, that way will lead to more monster encounters, so it is up to you which you prefer.
I hope this helps.