When you're ready to begin your ascent up Death Mountain, go to Kakariko Village and up to the guard at the closed gate. Equip the letter from Princess Zelda to one of the item slots, then "use" the item to show it to the guard, who will open the gate and let you pass. He also mentions that you should get a proper shield, which you should have in the Hylian Shield.
DEATH MOUNTAIN TRAIL
As you're going up the trail, you will encounter some Tektites, which do nothing but hop around; you take damage if you run into them. If you have the Hylian Shield equipped (and you should), you can guard with that to pretty much block their "attacks." You will also likely notice different colored walls, which can be blasted open with Bombs, which you will get in the next dungeon.
Continue up the path and you'll see a big boulder blocking the entrance, as well as a creature on the ground. Talking to him reveals he's a Goron and that a man in black armor blocked the entrance to Dodongo's Cavern. He says to go to Goron City to find the rest of them. Go up the path and you should see a red flag. Stay to the right of the path, being mindful of the rolling Goron who can hurt you, and continue into the cave and Goron City.
GORON CITY
Goron City is split into tiers, or floors, of which there are 4 of them. For now, head down to the very bottom and talk to the Goron by the stone door. He says that "Big Brother" shut himself in the room, waiting for the Royal Family's messenger. Stand on the mat in front of the door and play Zelda's Lullaby, which will make the door open.
Go inside and talk to Darunia, the boss of all the Gorons. He's angry because of everything that has happened. If you got Saria's Song, which you should if you're following this walkthrough, then stand in front of Darunia and play it to cheer him up. He won't hand you the Spiritual Stone of Fire unless you help him clear Dodongo's Cavern. He gives you the Goron's Bracelet, which lets you pick up Bomb Flowers.
Don't leave Goron City just yet, though. Use one of the torches in Darunia's chamber to light a Deku Stick, then light the four torches in the room outside, which will cause the giant urn to start spinning. Go to the floor above the bottom one and grab a bomb flower, throwing it in the urn when the smiling face is towards you. If done right, then it will stop on the smiling face and a Piece of Heart will pop out. It may take a few tries, but it's good to get it out of the way now. With bomb flowers, you can finally get into Dodongo's Cavern.
