After cheering up Darunia, you get the Goron Bracelets and can enter Dodongo's Cavern.
Get the Map
There are bomb flowers against the walls. Use a bomb flower to blow up the brown/cracked piece of wall straight ahead from the entrance. Go through.
Walk toward the center platform with the giant skull on it. Jump onto the nearby moving platform, then go from the Beamos platform to the platform on the left. Use a bomb flower on the brown/cracked piece of wall to find a treasure chest. Open it to get the map.
Get a Gold Skulltula
After getting the map, use the rising/falling platforms to go to the other side of the room. Take a bomb flower and run past the Beamos. Use the bomb flower to break the brown/cracked piece of wall. Go through.
Baby dodongos will come out of the ground and attack you. Once you attack them, they start flashing and will explode, so stay away.
Lure a baby dodongo to the brown/cracked wall ahead on the right and hit the Dodongo it to make it explode and break down the wall. If the Dodongos disappear, just walk around until more appear. In the room, kill the keese and gold skulltula with your slingshot, then do a jump attack to get the token.
Get the Compass
Back in the room with the baby dodongos, go to the right and pull/push the nearest statue onto the blue switch to open the door. Go through. Kill the keese and continue on to the next room.
Two lizalfos will attack you. One will crouch, then come near while the other jumps up and down and stays on a distant platform. When one comes near, swing at it with your sword once. If you get really close to it, it will jump over you and swing its sword, so do a backflip or just move backward to avoid it. After backing up, you can do a jump attack to hit the Lizalfos, or just walk closer and hit it with your sword. When they are gone, go to the next room.
If you don't have the Hylian Shield equipped, put it on now. Otherwise, if you get hit by fire, your Deku Shield will be burned and you will need to get another one.
If you need some Deku Sticks, use a bomb flower on the bombable wall that you pass by. There is a Business Scrub inside who will sell you some Deku Sticks.
To kill the dodongos in here, go in front of it until it stops and is about to breathe fire at you. Run around and hit its tail, then back up or do a backflip to avoid the flame on its tail that it will swing at you. In the 3DS version, it spins around twice to swing its tail at you. In the N64 version, it only spins around once. Keep attacking the dodongo this way until it starts flashing, then run away from it to avoid the explosion.
Kill the remaining dodongos and light a deku stick, then use it to light the three unlit torches in this room along the right wall, then go through the door that unlocks.
In the next room, step on the switch before walking too far forward. Once you step on it, jump down and go through the door that you just unlocked.
To get the compass, use a bomb on the bombable piece of wall on the left. The armos statue will come to life if you touch it or bomb it, and you have to kill it to leave the room. You can stun it with a Deku Nut and hit it with your sword, or you can try to get a bomb to explode on it. When you hit it, it will spin, then explode, so keep your distance.
Get a Gold Skulltula
Back in the last room, put a bomb flower down in the middle of the gap between the bomb flowers. This will make all of the bombs explode and make the stairs come down to your level. Go up and kill the gold skulltula on the webs up above.
Get the Bomb Bag
In the next room, equip the Hylian Shield if you have not yet done so. Use the slingshot to kill the fire keese, then walk around the platform, making sure not to touch any armos, until you see the ladder up to the top of the platform. There is an armos statue in front of it that doesn't come to life, so either pull it out of the way or just walk behind it and climb up. Step on the switch and go through the door that unlocked.
Run straight across the bridge, jump across the gap in the middle, and go through the door at the other end.
There are blade traps down below, so either climb down the ladder or jump into the gap beyond the ladder to avoid getting hurt. Approach the next passage to make another blade trap go past, then run straight through the next passage to the block in the wall while the blade trap is slowly returning to its starting point. Pull the block out of the wall. It will stop when it hits the platform behind you. The block has a recovery heart behind it that you can get if you wish. Otherwise, climb onto the block, then climb onto the platform that the block stopped on. Open the chest to get 20 rupees, then use the bomb flower to break down the bombable wall at the top of the ladder. It's possible to make the bomb land on the ledge, but it might be easier to just wait to throw the bomb such that it will explode in front of the wall. Climb up to the path that you opened and go through.
Use the slingshot on the eye switch straight ahead, then jump across and go through the arch, kill the dodongos (or ignore them), and follow the path to the door.
Kill the lizalfos in here like you did before. If you are low on hearts, go to the end of the long ledge to gather some. After the lizalfos are gone, go through the door that unlocked.
Shoot the eye switch above the arch, then jump to the next platform and turn left to shoot the next eye switch. Carefully jump to the far platform and go to the next room.
This is a room that you have been to before, with the blade traps on the lower level. Be sure not to jump down to there. Instead, jump straight across and open the chest to get the Bomb Bag. Keep in mind that these bombs seem to explode sooner than bomb flowers. Exit the room through the door to the left of the chest.
Create a Shortcut
Step on the switch up ahead to make one of the moving platforms rise up to this level, giving you a shortcut.
Get a Deku Shield
Go to the right and you will find a bridge. If you lost your Deku Shield, you can get another by bombing the wall on the other end of this bridge.
Get a Gold Skulltula
To access the next area of the dungeon, drop a bomb down each of the two gaps in the bridge. If you do it correctly, the two eyes of the dodongo skull will turn red and the mouth will open. Drop down one of the gaps in the bridge to land in an eye, then hop down to the beamos platform and go through the door in the Dodongo mouth.
Equip the Hylian Shield. Kill the fire keese with the slingshot, then drop down from the tunnel, climb the ledge to the right (across from the barred door) and go straight.
At the end of the tunnel, kill the fire keese, go through the room until you find the large blocks. Climb up the blocks to get onto the high ledge. Go to the far end of the walkway, jump off, then turn and face the wall that you jumped from. Push the block off of the edge, then move the block to to the bombable wall. Bomb it and go through to get a gold skulltula, then return to the last room.
Reach the Dungeon Boss
Kill the fire keese, then go up the blocks to get onto that long walkway again. Jump off at the end. If you have an empty bottle, break the two pots to reveal a fairy in one of them. Use the bottle near it to catch it, then unequip the bottle so that you don't accidentally use it. Go through the door here.
Go to the end of the tunnel, push the block off of the edge, kill the fire keese, then pull the block away from the wall and push it into the hole in the middle of the room. Go through the door that unlocks.
Drop a bomb onto the dark spot in the middle of the floor, then go past it and open the chest to get more bombs. Drop into the hole that breaks open when the bomb explodes.
Walk close to King Dodongo's face. When it opens its mouth, throw a bomb into it. When the bomb explodes, hit King Dodongo's face with your sword. When King Dodongo jumps up, walk onto the small slope down into the lava in the center. King Dodongo will roll away. Follow him. When he turns to face you, repeat the process of throwing a bomb, hitting it with the sword, then getting out of the way and following it. After you defeat King Dodongo, get the heart container and step into the light.
Your next destination is Zora's River, and there are some upgrades that you can pick up on the way.
Before that, if you go back to Goron Village, you can use a bomb on the Goron who rolls around an upper floor. If the bomb blows up on the Goron while he is in the tunnel in the back, he will give you a bomb bag upgrade when you talk to him.
You can also go back to Inside the Deku Tree and use your bombs on a wall in a room on floor B1 to get a Gold Skulltula.