Hi Emilio. To answer your question:
"Get up" can have 2 meanings: 1) to get out of bed, 2) to stand up with the intention to leave
"Stand up" does not mean to get out of bed, just to move from a sitting position to a standing position, but not necessarily leaving
"Wake up" is the process of your brain turning on as you exit sleep.
Examples in use:
In the morning, I wake up, then I get up and get ready to go to work. At work, I sit all day, so I like to stand up every 30 minutes in order to stretch.
Hope that helps!