This is the dungeon where you start. What follows is a quickest route out. The dungeon isn't very large, so feel free to explore. The cave where you start is a safe place to rest until you leave it for the first time. After that it is subject to random encounters.
From your cave, go down the corridor and through the secret door at the end. Kill or avoid the rat, (if you want to join the Thieves Guild, this is a very good opportunity to pickpocket in order to gain their attention. If the rat inflicts too much damage on you, you can always go back to the cave and rest. You might have a random encounter, but it's relatively safe. There is a treasure pile on the table in the corner. Collect it. When you are ready, go through the opening and up the stairs to the top.
Follow the corridor around (kill the bat). The first door on your left has an imp and some treasure behind it. The imp can sometimes be a very tough customer for a low-level character because of his spell ability.
Because imps have hands, they can also open doors. Save the game before messing with the imp. If you are lucky, you'll take a little damage from his spell and can then go rest. If you aren't so lucky, you'll need that saved game for another shot. The second door on your left has a rat and no treasure behind it.
The next opening on your left leads down to a room with a Thief and some treasure. The Thief isn't too terribly tough, but he is a fast attacker. This is another place where you should save the game. When you get to the stairs going up (and kill the bat), you're ready for the next section.
Climb the stairs and go through the door. Kill the Archer (or run very fast so he doesn't shoot you in the back) and go out the door to your left. There are two rats in the room behind the archer and a steep incline which takes you to the rooms at the top of the big room. In these rooms you will find a bear (no treasure), a thief (treasure), an imp (treasure behind the bookcase), an orc (no treasure) and a room with treasure, but no critter. After going through the door, you'll be in a large chamber with a wide stairway going up in the middle. (Behind the door at the back of this room is a bear.) Kill or avoid the bat and climb the stairs. Kill or avoid the Skeletal Warrior at the top of the stairs. There will be a throne on a dais against the wall. Jump up on it and pull the lever next to it. The dais will go up, giving you access to the opening above. Through the door to the left of the stairs at the bottom, you will find a section with a bat, a rat and two archers. The stairs at the back of that area will take you up to the landing above the throne.
Don't go through the door (it leads down the archers, rat and bat -- see above). Instead, follow the corridor. At the top of the first incline will be a rat. Go up and go through the first door on the right. There will be an imp, a bat and a rat in the room. Kill or avoid them and click on the archway on the right wall to exit. If you feel like exploring the rest of this area, you'll find a rat farther down the hallway, a skeletal warrior in a room off to the side, and a thief and another skeletal warrior in the room at the end.
You have completed your first (of many) dungeons. The image to the right is what all exits look like. There will be many opportunities for you to appear in a dungeon and have to find your way out again, so this is what you are looking for. This is almost always a welcome sight, so enjoy it.