Go the Lumbridge Swamp and make sure you have your lantern lit before you enter. Use your rope on the hole, then put your Spiked Helmet on and walk southeast of the caves.
Watch out for Wall Beasts! They will not hurt you if you are wearing your helmet!
Use the stepping stone and walk in the cave. Walk west and climb down the rock. Follow the path and you will see Juna. Walk to her and she will start the quest. She will tell you that you need to mine some rock.
The Sapphire Lantern
Go back outside of the Slayer Dungeon and to Lumbridge Swamp. Put on your lantern. (If you do not put it on while in the cave, you will not be able to see and might die!) Cut your sapphire, if you had not already. Use your cut sapphire on your lantern and light it again.
The Light Creatures
Go back to the cave and talk to Juna. You will see some light creatures floating around. Click on your lantern and examine one of the light creatures. It should pick you up and carry you across the gate.
Mine one of the rocks in front of you.
When you have done so, use your chisel on the rock to make a bowl. There will be some more rocks near you. Go down them and talk to Juna again. She will now let you use the mini game.