If the cave is down, your first is correct. Then the second should be "up out of the cave".
Student or Learner
I am near a cave and I want to visit it. So I am going to go down into the cave. Is it right? And what am I going to do next? Am I going to go up from the cave?
Please, show me every single one of my mistakes. Thank you.
I would go down into the cave and then come back up from the cave.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.