Death Mountain Trail:
Head to Kakariko Village and then up Death Mountain Trail. You'll need to get past the guard by showing him Zelda's Letter (given by Zelda). Meet Darunia in Goron City. Play Zelda's Lullaby to open the path to Dodongo's Cavern.
Dodongo's Cavern:
This dungeon requires you to use bombs to open paths and defeat enemies. Obtain the Bomb Bag. Push blocks onto switches and light torches. The boss is King Dodongo. Roll a bomb into his mouth when he inhales to stun him, then attack with your sword. After defeating him, Darunia will give you the Goron's Ruby.