Dungeon 2 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.
Get a Key
You don't actually need any keys to get through this dungeon. Skip ahead on this page if you want to go the quick way. But if you want to get keys anyway, go north from the entrance and kill all the Ropes (snakes) to make the west door open, then go west and get the key, killing the ropes if you feel like it.
Get a Second Key
From the previous key, go east twice, then go south. Kill the ropes to make a key appear.
Get the Compass
Go north, then east through the locked door. Get the compass from the corner of the room.
Get the Map
If you have bombs, put a bomb in the middle of the north wall and go through the hole to get the Map. Otherwise, go west, then north, then through the locked east door to get the map.
Get the Magic Boomerang and a Third Key
If you have bombs, put a bomb in the middle of the north wall and go through the hole to reach a room with three Blue Goriyas. Kill them to make the Magic Boomerang appear. It does the same thing as the regular boomerang but flies farther.
From the Magic Boomerang room, go west to a room full of ropes. Kill them to make a key appear.
If you don't have bombs, go through the west door from where you got the map, then go north to a room full of ropes. Kill them to make a key appear.
Then go through the east door to reach a room with three Blue Goriyas. Kill them to make the Magic Boomerang appear. It does the same thing as the regular boomerang but flies farther.
Get a Fourth Key
From the Magic Boomerang room, go west and then north to a room filled with sand. Two Moldorms will attack. Just hit their segments until they die. Take the key that appears.
Optional: Get Bombs
Go east from the Moldorms room to find some bombs. You can optionally put one in the middle of the north wall and go through the hole and kill all the Gels to get a blue rupee.
Get More Bombs
If you're in the room with sand, go north. There are a bunch of ropes in here. You can go through the locked east door and kill all the Gels in the next room to get a blue rupee. From there, you can go north to get another famous hint from the old man: Dodongo dislikes smoke.
Otherwise, just go north from the sand room until you reach a room with five Red Goriyas. Kill them all to make bombs appear in the corner. Killing them also makes the north door open. Take the bombs and go north, then skip ahead to the Dodongo section below.
The Quick Way
If you don't want to bother with collecting keys in this dungeon, go east from the entrance, then go north until you reach a room with four blocks. From that room, go east and kill the blue Goriyas to get the Magic Boomerang. Then go west again, and finally go north to the dungeon boss's room, killing enemies if the doors are closed on the way.
Now you have arrived at the boss of dungeon 2: Dodongo. You need bombs to defeat him. Put a bomb in the path of his mouth when he is walking toward you, and he will eat it, and then it will explode in his stomach. Do this twice to defeat him. Get the Heart Container that he drops, then go west to get the second Triforce piece.
The next dungeon is in the southern part of the map, three screens west of where you began the game, although you can't get there directly. You have to go north of where you started, then go west three screens, then go south, then east.
To get there from dungeon 2, go south, east, south, then west until you can't go west anymore. From there, go south, cross the bridge and go west, then go west again, then south, then east.
The next dungeon is Dungeon 3: Manji.