Starts with F1-F6 displayed in a list firstly in random order either on display or serial print
How? The values need to be stored in some sort of list, queue, stack, or array. You can't manipulate what has been displayed in the serial monitor.
Then with the 6 inputs as the inputs (F1-F6) the first time it is pressed it takes that F number off the list and the second time it is pressed that F number appears on bottom:
What are the inputs? Switches that people press? Switches that machines press?
F4 gets job
What job? How?
Keeping an array with values in it, removing values from the array, adding values at the end of the array, etc. are all relatively simple tasks.
What code do you have so far?