Goron City:

Go to the bottom level and stand on the doormat in front of the closed door and play Zelda's Lullaby.  Then go inside and talk to the Goron King.  Then play Saria's Song for him.  He'll give you the Goron's bracelet.  Now leave Goron's City and head up Death Mountain trail and take the first right you see.  Pick up the bomb plant there and chuck it down to the big boulder below.  If you aim right you'll blow it up.  Then go around and enter the revealed cave, which is Dodonogo's Cavern.

Dodonogo's Cavern:

WARNING about this level:  if you have your Deku shield equipped and you get burned by a fire bat or something, the shield will burn away!  So use the Hylian shield unless you need your Deku shield to fight lizardmen or bush guys.  You can buy another shield from a bush guy on the left side of the Big Room.

In the first room, use one of the bomb plants to blow open the doorway.  Then head into what I will from now on refer to as the Big Room.  Make your way to the right side and use the provided bomb plant to blow up the doorways and that laser-shooting thing.  Take the doorway that leads to a passageway (the other just leads to a gossip rock) and kill the first couple of lizards.  Watch out because when these lizards die they explode.  On the right wall near the back of the passage is a doorway but you have no bomb plant to blow it up with...kill one of the lizards next to the doorway and let its explosion as it dies open the doorway for you.  This room has some bats and a gold skulltula in it.  Now move on to the end of the passageway and drag a statue onto the switch to open the door.  Go through the next passage and then you'll find yourself in the lava room, facing two deadly lizardmen.  These guys are tough, Z target them and use your Deku shield to block and counter with slashes or jump slashes.  You can also shoot them with the slingshot but that's usually more trouble than it's worth.

After you kill the two lizards go through the new door that just opened.  Notice the bombable doorway down this next passage.  Go on a bit further and you'll see a bomb plant you can use to blow it up.  There's nothing really useful in that room though.  Move on down the corridor and kill those lizard things by hitting their tails (watch out they blow up too). Use a Deku stick to light all the torches in the room and open the door.  You'll now be back in the Big Room.  Step on the switch in front of you to unlock a door across the way.  Now head to the other side of the room and find the door that was just unlocked but don't go through it yet.  First blow up the doorway next to it and get the dungeon map.  Now go through the door.