After you have rescued the dwarven swordsmith and rescued the maidens in the Ice Palace and Misery Mire, some new upgrades become available.
Get the Super Bomb
Use the Magic Mirror to go to the Light World, then play the Flute and go to 4, your house. Go north to Hyrule Castle and walk into the castle gate to be sent back to the Dark World, then go east, then south to the bridge, use the Hammer to pound the posts into the ground, and go west to where your house is in the Light World. Go inside.
You will see a large pink bomb for sale. This is the Super Bomb. Purchase it for 100 rupees, and it will follow you until you press A or jump down from a ledge, at which point a 3-second timer will be displayed on screen. After the timer counts down, the bomb will explode. Be careful not to press A or jump down from a ledge until you are ready to detonate the bomb.
Use the Super Bomb at the Pyramid of Power
Your next destination is the Pyramid of Power. To reach it quickly, go outside and use the Magic Mirror to return to the Light World, then go down the slope in front of your house, and go north from there to Hyrule Castle, and go through the castle gate to return to the Dark World. Then go north up the pyramid's stairs and go west to the cracked wall. Make sure that the Super Bomb is in front of it, then press A and move a short distance away to avoid the explosion. Afterward, go into the hole that you opened.
Receive Upgrades from the Cursed Fairy
Approach the small pool and you will be asked to throw something in. First, throw in the Bow and Arrows to get the Silver Arrows. You will need these to defeat the final boss.
Then approach the pool again, and this time, throw in your sword. You will receive the Golden Sword, which is twice as powerful as the Tempered Sword.
You can also throw in an empty bottle to get Medicine of Magic if you wish. You may want to fill two bottles with Medicine of Magic before you go to the next dungeon.
Open the Next Dungeon
Go outside and use the Magic Mirror to go to the Light World, then play the Flute and go to 1, Death Mountain. Go east to the old man's cave, go in, talk to the old man if you need to heal up, then go up the stairs behind him and go through the east door. Go east again, then south to go outside. Go west to the ladder and climb up. Go east and step on the tile to go to the Dark World. Go west and stand on the outline of Spectacle Rock and use the Magic Mirror to go to the top of Spectacle Rock in the Light World. Jump off of the north edge and go east past the Tower of Hera to the next area.
Cross the bridge, then use the Hammer to pound down the posts in the way. Go east from there, past the caves, to reach the next area. Go to the south side of the hill in the middle of the area. Lift and throw the large black boulder, then go up the ladder. You need to use the Hammer on the posts in the correct order. First, hit the one on the right, then the one up above, then the one on the left. This will make a warp tile appear. Step on it to go to the Dark World.
Stand on the symbol on the ground and use the Quake Medallion. The dungeon will open.
Get a Piece of Heart
Before you go into the dungeon, jump down from Turtle Rock and go west. Pick up and throw the large white boulder and go down the stairs.
The chests throughout this room just contain rupees. If you wish, you can use the Hookshot to reach them. Otherwise, from the entrance, go north to the end of the path. There is an invisible path north of you. You can use the Ether Medallion to reveal the path, but it costs a lot of magic and you might not be able to remember the exact path as you go across. Another method is to use the Cane of Somaria. Use it to create a block north of you, and slowly push it north until it falls off of the edge of the invisible platform. Then create a block east of you, and push it east until it falls off of the edge. Create another block north of you, push it north until it falls. At that point, go west until there is a visible platform north of you, and walk north onto it. Use a bomb at the cracked wall and go through.
In this plus-shaped room, you can optionally bomb the north wall to find a fairy fountain. The east wall doesn't open if you place a bomb there. Bomb the west wall and go through.
You can optionally bomb the north wall in this hallway to reach a room with a few small fairies flying around. Then go south to go outside. Use the Magic Mirror to go to the Light World, then take the Piece of Heart that you find there.
Step back into the sparkling spot to return to the Dark World. Jump down from the ledge and you will reach the nearby part of the mountain. Go east from there to find Turtle Rock. Pull on its foot (the one on the right side of the screen) to get 20 rupees, then enter Turtle Rock.