You are awoken by a cry for help from Princess Zelda, who is being held prisoner in the castle.
Help Zelda
After your uncle departs, you must go against your his orders and attempt to rescue Zelda.
Open the chest in the corner (press A in front of it) to get the Lamp. (If you don't get it now, you have two more opportunities to get it.) The lamp lights up dark rooms automatically. Later, when you obtain magic, you can use the lamp to light torches.
Go outside. You can pick up bushes with A. Some of them might have rupees, which you can use to buy things later.
Go north from your house and walk off of the back ledge, then exit to the north. Go west until you see a bridge to the north. Go north up the bridge until you can go east. Follow the stone path until you find a bush with a stone border around it. Lift it to reveal a hole. Fall into it.
Secret Passage
From where you land, go west and talk to your uncle. You will get a sword and shield. You can swing the sword by pressing B, and charge up a spin attack by holding B, then releasing it.
Go west, then south and through the door. Go east and kill the soldiers, then continue going east to find a chest. Open it to get a blue rupee (worth 5 rupees), unless you didn't get the Lamp in your house, in which case the chest will hold the Lamp. Be sure to lift the pots to get a couple of small magic jars. Now that you have partially filled your magic gauge, you can press Y to use the Lamp, but this will drain your magic gauge by a small amount.
Go west again, then go south up the stairs and go through the door at the top.
Hyrule Castle Courtyard
Cut the bushes, then go west and cut through the bushes there to reach the central walkway. Go north and fight the green soldiers as needed. One of the soldiers has a larger sword that your sword will bounce off of. Try to hit the soldier's shield arm to avoid this. Go north through the door to enter the castle.
Hyrule Castle
Go east from the entrance and go through the door there. Follow the carpet to a door in the north wall, fighting the green soldier on the way if needed. In the next narrow hallway, there are three green soldiers, one with a big sword. After you get past them, go through the northwest door.
Continue going west, until you see some stairs going down. Go down them. You will see a blue soldier with a big sword. Kill it and it drops a small key. Take it, then open the chest here to get the dungeon map. You can press X to view it. Go through the locked door below.
In the next area, avoid falling into the pit. Go down to the lower level and go south past the blue soldier, then exit to the south.
Go south until you can go no farther south, then go west to the next area.
To avoid the green soldiers here, go up the stairs, then go east until you can go south down a narrow path. When you can go no farther south, go west, then go north down the stairs and continue going north to reach the next room.
Kill the green soldier to open the doors, then go through the east door.
Kill the blue soldier to get a key, then open the chest to get the Boomerang. Press Start, and use the directional pad to equip the boomerang. You can throw it at enemies to stun them, and throw it at items (such as rupees and hearts) to bring them to you. Go west to the previous room.
Go north up the stairs and through the locked door. Follow the path until you arrive in the jail. Kill the green soldier. The pots in open the jail cell contain hearts, but don't take them unless you need them.
Go east from the open jail cell and you will find a ball and chain soldier. You can try picking up the pots from the open jail cell and throwing them at the soldier. Two pots should kill it. Otherwise, quickly run up to the soldier and hit it before it spins its ball and chain, then run away quickly to avoid getting hit.
The soldier will drop the big key. Take it walk up through the door of Zelda's cell, and talk to her. Open the chest next to her to get a blue rupee (or the Lamp if you still didn't get it). Then you must help Zelda escape from Hyrule Castle.