After you complete dungeon 1 in the second quest, you can get an additional heart container and get the White Sword.
Get a Piece of Heart
Go to the southwest part of the map. There is a red forest there where the ground is in a + shape. When you reach this area, go north, west, south, west. You will emerge in a new area of the forest.
Go north four times and push the grave indicated by the X below:
O O O O O X O O O O O O
Stairs will appear. Go down those stairs and take the heart container.
Get the Power Bracelet
Go south, then east twice. Go north and enter the building to get 30 rupees.
Go back outside, then go south, east, and northeast. Then go east and touch the northeast statue to make it move. Get the Power Bracelet from the place where the statue was standing.
Get the White Sword
Go west and south, then northwest. Push the leftmost block and go down the stairs. Use the rightmost stairs.
Go west three times, and go up the stairs. Go into the cave and get the White Sword.
Go to Dungeon 2
Go outside and go south, then if you need hearts, go east, then go north until you reach the fairy fountain. From there, go south. Then go west until you can't go west anymore. Then follow the water's edge until you reach a short bridge. Go over it, then go west, then go north until you can't go north anymore. Touch the top middle statue to reveal stairs. Go down to reach Dungeon 2.