journeymen,
With a double set trigger that is properly adjusted, you "set" it by pulling the rear trigger until it "clicks". Then, to fire the rifle( with the safety off), you only have to "touch" the front trigger. If the safety is "on" the set trigger will trip if you touch it, but the rifle shouldn't fire. Since the safety is a mechanical device and might not work every time, the trigger should not be set until right before you shoot. If, for some reason, you don't take the shot you should open the rifle and trip the trigger. With practice, you can hold the rear trigger back and "ease" it forward; but you should still open the rifle first. With a double set trigger, there is no reason to push either one forward. However, some guns( such as combo guns and some single shots) have a single set trigger and you do push them forward to set them. Other than the different way to set them, they should be treated the same as double set triggers. Any set trigger is a separate mechanism that works independently of the safety or the mainspring/sear/hammer. You can fire the rifle without setting the trigger, by pulling the front one, but the "trigger pull" will be very poor.
Mike