Break out of your cell by pushing the door, then handle the guard.
The Gold Key is in the cell to the left.
There is a secret corridor behind the cells that takes you to the Silver Key,
but you may also get some guards to open the silver key door for you once you enter the main hallway.

Make it past the SS behind the green barrels then head on towards the elevator.
You won't get a machine gun though unless you move around towards the elevator to the secret level, past the silver key door and a secret door. If you do don't forget to fetch the chaingun and the 1-up as well before moving up to the secret floor.