After your visit to Kakariko Village, show Zelda's Letter to the soldier and go to Death Mountain Trail.
Climb the Mountain
You will encounter tektites along the path. Don't try to stab them with a forward thrust, because they will likely just jump over your sword. Instead, target them and swing your sword without touching the control stick.
You will see a pile of rocks with a Goron in front of it, but you can't do anything about it, so just continue going up the mountain. When you see a flag on a wooden pole, go to the right of it and go up the slope there.
As you go up the mountain, a Goron will roll toward you. It will hurt you if it hits you, so stay close to the left wall to avoid it.
From there, go straight ahead past a platform with small rocks on it. Go through the cave entrance beyond the platform to reach Goron Village.
Get a New Item
Go down to the bottom floor and stand on the carpet next to the closed door. Play Zelda's Lullaby to open the door. Go through and stand near the angry-looking Goron and play Saria's Song to cheer him up. He will give you the Goron Bracelet and ask you to get rid of the dodongos in Dodongo's Cavern.
Get a Piece of Heart
Before leaving Darunia's room, take out a Deku Stick and light it with one of the torches near Darunia. Leave his room and light the four torches here on the bottom floor. The giant jar will start spinning.
Go up to the next floor. If you throw a bomb flower into the jar when the happiest face is looking toward you, you will get a piece of heart. There is a bomb flower in the hallway near the jar on the second floor. If you are having too much trouble, you can come back later when you have an item that makes it easier (you'll know it when you get it).
Open Dodongo's Cavern
Go up to the top floor of Goron Village, then go outside to Death Mountain Trail and turn right. There is a big Goron standing near a bomb flower. Use the bomb flower to break open the rocks that are blocking Dodongo's Cavern. Just pick up the bomb flower and throw it over the short piece of fence. If you didn't aim properly, just wait for another bomb flower to grow, and try again.
Get another Piece of Heart
Before going down to the entrance, you can get a piece of heart that is on the ledge above the entrance to Dodongo's Cavern. To reach it now, stand with your back against the short piece of fence next to the bomb flower. Pick up the bomb flower and throw it, then do a backflip toward the left. If you aimed correctly, you will land on the ledge above the entrance to Dodongo's Cavern. You will get hurt, but when you get the piece of heart that is here, your health will be refilled.
Get a Gold Skulltula
Go down to the entrance to Dodongo's Cavern. If you have bugs in a jar, drop them into the spot of soft soil here to get a gold skulltula.
Go into Dodongo's Cavern.