Before beginning this quest, you should buy an appropriate level weapon for yourself. These slimes have a lot of health and using the training weapon will take a lot of time. Slime Cave is located one screen east of the entrance to Bluevale Cave. After receiving the quest from Folmar, your character will recognize that this is the entrance that Folmar described as you get close to the entrance. Simply interact with the mine shaft and you will have entered Slime Cave.
In the second image, the switch solution is provided to you. There is a slip of paper with a hint (basically the solution) in the lower right hand part of the room. Also notice that the Putrid Slime monster is located behind the lowered pillars. Make sure to kill at least one to complete the quest.