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After you defeat the King Dodongo and get the Goron Ruby, your next destination is Zora's River. You can get some upgrades on the way.
After getting the Goron Ruby, go up to the part of the mountain where your path is blocked by boulders, to the left of the red flag. Drop and throw bombs to get rid of all of the boulders. If you run out of bombs, pick up and throw the rocks near the Goron Village entrance, or just go into town and buy more from the shop on the bottom floor.
After you bomb all of the boulders, equip the Hylian Shield and follow the path up the mountain. When the explosions start, press R to hide under the shield. Run forward after some rocks land on your shield, but as soon as you see a shadow over you, hide under the shield again. Eventually you will reach a wall that you can climb.
Kill the skulltulas on the wall with the slingshot, then climb up. When you reach the first ledge, use the slingshot to kill the skulltula near the top.
Don't talk to the owl up above yet. Instead, bomb the rock in the wall and go inside. Play Zelda's Lullaby on the Triforce symbol. The fairy will teach you a spin attack and give you a magic meter. Go back outside.
Before talking to the owl, go into the cave entrance. It's very hot inside, so you can't stay for long. Just do a roll attack on the crate to find a gold skulltula.
Talk to the owl at the top of the mountain and then stand below its shadow to get carried back to Kakariko Village.
Don't jump off of the roof yet. Instead, turn right and go to the right side of the edge that you are facing. Down below you will see the graveyard entrance. The roof of the cucco pen is below you. Land on the roof and go into the crack in the wall to get a piece of heart next to a cow. Go back outside.
The entrance to Zora's River is between the entrance to Kakariko and the entrance to Kokiri Forest. Turn left from the stairs to Kakariko, then stay along the left wall in the bank of the stream.
Walk toward the boulders to hear from the owl. Then roll attack the tree to get a gold skulltula.
Bomb the boulders that block your path into Zora's River. Stay on this side of the river and go as far as you can to find a guy eating beans. He will sell you Magic Beans. You can get a total of 10 magic beans from him. The total cost is 550 rupees. The price for the first set of beans is 10 rupees, then the next set is 20, then the next is 30, and so on. Buy as many beans as you can afford. Plant one bean in the soft soil next to the bean salesman. This soft soil doesn't have a gold skulltula inside.
Go back to the cucco that you passed while walking to the bean salesman. Pick up the cucco and jump across at the bend in the river. Follow the path. You will jump across the water near a narrow gap in the wall. After that, you will see a hill that slopes upward. Still holding the cucco, go up the slope and jump across the gap there. You will be on an upper ledge. Jump toward the high ledge up ahead and then throw the cucco just before you bump into the ledge. You should catch the ledge and be able to climb up. From there, jump to the platform with the piece of heart.
Go to a log in the river and stand on the light part of the log. When you notice the frogs looking at you, if you have room in your wallet for 50 rupees, take out your ocarina and play any song you know. You will get 50 rupees for each song, and one of the frogs will grow larger. If you run out of room in your wallet, go and buy some more magic beans.
Go back and get the cucco again. Take it all the way to the waterfall at the end of the river. There is a platform that you can reach using the cucco to get another piece of heart.
Find the ladder that goes up from the river near the base of the waterfall. Use Sun's Song to make it night, then kill the gold skulltula that appears on the ladder.
Go to the ledge facing the waterfall and stand on the tile on the ground that has writing on it. Play Zelda's Lullaby to open the waterfall. Jump into the passageway and go in.