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.
Get a Key
Go west from the entrance and get the key.
Get a Second Key
Go north and kill the Zols to get another key.
Get Some Bombs
Go north. If you already have eight bombs, use one on the middle of the east wall. Then kill the Darknuts to get some more bombs (they are only vulnerable from the side or back). If you didn't have any bombs before, use one on the middle of the east wall now.
Optional: Get Five Rupees
Go east through the hole that you made in the wall, then kill all the Darknuts to open the east door. Kill the Keese and Zols in this room to make a Blue Rupee appear. Then go west again.
Get the Map
Go east through the hole that you made in the wall (if you didn't already), then go north from the room with the three Darknuts. Avoid the spike traps and get the Map.
About the Dungeon's Shape
After you get the map, you might be wondering why the dungeon is in the shape of a symbol that many find offensive. First of all, this symbol wasn't considered offensive until World War II, which began in 1939. But this game was released in 1986, so it had a negative connotation at that point, right? Well, this symbol has actually been used as a sacred symbol of good luck and protection from evil in Hinduism, Buddhism, and other religions and cultures since at least 200 BC. Buddhism is widespread in Japan (where The Legend of Zelda was made), and this symbol is still strongly associated with Buddhism there. So strongly, in fact, that you can even see this symbol on Google Maps right now as a symbol for Buddhist Temples. Here is an example: Buddhist Temples in Japan.
So to make a long story short, this symbol was associated with Buddhism in Japan since long before World War II. It's not meant to be offensive here.
Get a Third Key
Go south from the room where you got the Map, then go west, then north. Kill the Zols here to get another key.
Get the Compass
Go south, then west, and avoid the spike traps to get the Compass.
Get the Raft
Go west through the locked door, then kill the Darknuts to open the south door. Go through it. You can optionally kill the Darknuts in here to get more bombs. Then go down the stairs. Use the ladders to reach the Raft. You can use this in the overworld at certain docks to go across the water.
Get a Fourth Key
Go back upstairs, then go north twice. Get the key from this room.
Get More Bombs
Go east and kill the Keese to make bombs appear in the middle of the narrow passage. Then exit through the eastern locked door.
Optional: Get Fifth Key
Go north through the locked door here. Don't enter the next room too quickly, because there are spike traps along the walls. Kill the Zols, then push the block on the left. This will open the north door. Go through it.
The Old Man will ask if you got the sword from the Old Man at the top of the waterfall. If you have been following this walkthrough, then yes, you did. But if not, you can later go and get it if you have five heart containers. The waterfall is in the northern part of the map, slightly east of center.
Anyway, go west from this old man and kill all the Keese to get another key.
Go to the Boss
The dungeon boss is in the middle of the far eastern rooms of this dungeon. To get there from where you got the key in the previous paragraph, go east, then south until you are in a room with a bunch of blocks in a backward-C shape, then go east through the locked door, then use a bomb in the middle of the east wall. Go through the hole to reach the boss.
Manhandla has four mouths, each of which shoots fireballs. You have to kill all four mouths to defeat Manhandla. A bomb will kill a mouth in one shot, and if you're lucky, you can place the bomb such that it will explode in the middle of him, killing all four mouths in one shot. Otherwise, you can use your sword or other weapons to damage the mouths until Manhandla dies.
Take the heart container that Manhandla leaves behind, then go north to get the third Triforce piece.
Now you can go to Dungeon 4: Snake.