After you get some upgrades to prepare for the first dungeon, you enter Dungeon 1: Eagle.
Get a Key
From the entrance, go into the east door. Kill the Stalfos that has a key inside, then get the key. You can kill the other Stalfos if you feel like it.
Get a Second Key
Go west twice to reach a room with Keese in it. Kill the Keese to make a key appear. Take it, then go east.
Get a Third Key
Go north through the locked door. The next room has some Stalfos. You can kill them or just ignore them, and go through the north door.
Kill all the Stalfos in this room to make a key appear.
Get the Compass
Go east if you want the compass. It shows you which room in this dungeon has the hidden Triforce piece in it. Then go west to the previous room.
Get the Map
If you have bombs, you can bomb the middle of the north wall and go through the hole to reach the room where you can get the map. Otherwise, go west into a room full of Keese, then unlock the north door and go through. In the next room, if you want a meaningless but famous hint, kill the three Zols and then push the block that is in the middle of the room. The west door will open and you can go through to be told "Eastmost penninsula is the secret." Nobody is really sure what the eastmost peninsula is. The original Japanese hint in this room says that you can't fire arrows if you have no rupees.
Anyway, from the room with the three zols and the block in the center of the room, go east to reach a room full of Zols and the map.
Get a Fourth Key
Go north to a room with a narrow path surrounded by water. Some Stalfos are walking around in here, and one is holding a key. Kill that Stalfos and get the key, then go north through the locked door.
Get a Fifth Key
Kill the three Red Goriyas in here to make a key appear.
Get the Bow
Go west through the locked door to reach the next room, but don't enter the room quickly. There are some spiked traps that will come toward you when you approach. Wait for them to move back to their original places before running past.
Go to the left side of the room, but be careful to avoid the traps up there. Then push up or down on the leftmost block and go down the stairs. Climb down the ladder, then go to the right and climb the ladder there, then walk to the Bow to get it. Unfortunately, if you have no arrows, you can't use the Bow. Indeed, you can't even select it in your inventory. But you can buy some arrows later. If you do have arrows, keep in mind that it costs a rupee each time you shoot an arrow.
Get the Boomerang
Go back up the ladder to the room with spike traps. Go east from this room, then go south to the next room. Go south one more time. If you want to avoid using your key and you have some bombs, go south, then east, then bomb the middle of the north wall and go through the hole. Otherwise, go east through the locked door. Kill the three Red Goriyas in here to make the Boomerang appear. You can throw the boomerang and it will come back to you. When the boomerang hits most enemies, it stuns them. You can also use throw the boomerang at certain types of items (such as hearts and rupees) to collect them.
Get a Sixth Key
Go east from the room where you got the boomerang. Stay away from the walls, because giant blue hands called Wallmasters will come out and try to grab you. If one of them catches you, it will send you back to the first room of the dungeon.
Get the key from this room if you need it.
Go north and you will be in the room of the dungeon boss: Aquamentus. It spits three fireballs toward the left of the room repeatedly. Its only vulnerable spot is its horn, but it can be difficult to hit it without getting injured. If you have bombs, just put a bomb near its head to damage it. You can also use arrows if you have any. Aquamentus is also vulnerable to sword beams.
If you are having too much trouble, you can get the Magical Shield (which blocks fireballs) for 90 rupees if you leave the dungeon, go east across the bridge, then go south, west, and west again, then use the Blue Candle to burn the corner tree.
If you want arrows instead, leave the dungeon, go east across the bridge, then go south twice, east, north, and east, and go into the cave there to buy the arrows for 80 rupees.
Get a Heart Container
After you defeat Aquamentus, it drops a Heart Container.
Get the Triforce Piece
Go east from Aquamentus's room and get the Triforce Piece. You will automatically be transported out of the dungeon.
From here, you can explore Hyrule if you wish. As mentioned in It's Dangerous To Go Alone, you can get three heart containers, the Blue Candle, the White Sword, and the Letter to the Old Woman if you feel like it.
You can also buy a blue or red potion from the old woman if you have the letter. The red potion fills all your hearts, then turns into a blue potion. The blue potion fills your hearts and then disappears. The nearest potion shop is in the screen north of the dungeon, but you will have to go east, north, and west, to get there. Then bomb the wall on the right side to reveal the entrance.
You can also buy arrows from a couple of shops for 80 rupees. There is also the Magical Shield, which blocks fireballs, which you can buy from a hidden shop for 90 rupees. Read the middle paragraph in the Aquamentus section above for directions to that shop.
After you have made whatever preparations you feel like making, you should go to dungeon 2, which is five screens east of dungeon 1, but you can't go directly east to get there. If you're at dungeon 1, go east, then north, then east five screens, then south twice, then west, then north. There, you can enter Dungeon 2: Moon.