You use 'read myVar' keyword to store the input from stdin (keyboard) into variable myVar. So you just need to read the input for each case. Now when you reference variable, that is when you are retrieving the information stored behind it, you use dollar sign, which you were missing in touch and rm commands.
#!/bin/bash
#Sample Script
VAR1=
echo "Choose an option: 1) Create a file. 2) Delete a file. 3) Move a file."
read VAR1
case $VAR1 in
1)
echo "Pick new file name."
read VAR2
touch $VAR2
;;
2)
echo "Pick name of file to delete:"
read VAR2
rm $VAR2
;;
3)
echo "Select Source File:"
read VAR2
echo "Select Destination:"
read VAR3
echo "Source is $VAR2 and destination $VAR3"
;;
*)
echo "Invalid option, please try again."
;;
esac
Huh? Not sure what you're getting at here. All the world can usually execute the 'rm' command. Whether it's run from a script or an interactive prompt, the user still needs to have the appropriate privilege.