Things You'll Need
Instructions
Prepare yourself for a mining expedition. Create several iron or stone pick axes, two stacks of torches and an iron or diamond sword. Take a few pork chops and equip a suit of armor for extra protection against large Slimes.
Find a suitable area to start digging. It should be enclosed and well lit so hostile monsters, like Zombies or Creepers, will not spawn on top of you while you are mining. Begin to dig downward in a spiral pattern. Create stairs with blocks as you dig. This will keep you from digging straight downward and will provide a way to ascend out of the mine when you finish.
Dig downward until you hit bedrock blocks. Bedrock blocks indicate that you are in the bottom 16 layers of the map. When you hit bedrock blocks, do not continue digging downward. If you haven't found a cave system containing Slimes at this point, you can create a cave on your own in which the Slimes can spawn.
Mine out a room at least five blocks high and nine blocks wide. Cover the walls in torches and wait for Slimes to spawn. If the room is big enough and well lit, they should eventually appear. Use your iron or diamond sword to protect yourself against larger Slimes.