Go up and to the left. The cave contains a thief with 300 rupees. In the room behind him is a Piece of Heart. Go a screen to the left into the desert and go straight up and into the cave. Light the torches and get a Piece of Heart. Now warp back to Link's house and head a screen left and down to the lake.
Go inside the cave marked with shields for a Great Fairy and 300 rupees. Go down beside the Fortune Teller's shop and hookshot to the right to get a Piece of Heart. Fall in the water, swim to the right, and visit the greatly changed Fairy Upgrade Island for a thief with 300 rupees. As if that wasn't enough, bomb the wall behind him for another 300 rupees! Go back outside.
If it's after go north-east from here and save the Priest, who has fallen in the water. You will be taking him to Hyrule Castle, but first swim back up the river in the direction he came from, until you reach the bridge in front of Hyrule Castle. Go under the bridge to get a Magic Bottle from the man just like you do in A Link to the Past. Stand on the shallows, call the duck, and choose a nearby point, then go all the way back to the castle. Then go inside walking in a straight line until you find Zelda and the fortune-teller, at which point the Priest detaches from you and rewards you with 10,000 points.
Now go back to the desert and enter the monstrous mouth.