How is this accomplished? I've heard something about replacing ALL of the factory speaker wire?

There are many ways, and it depends on the direction you are going to take. Are you buying a new amplifier and head unit?

Right now I am just buying a new head unit

Right now I am just buying a new head unit

run new wires to the speakers.

Is running new wires to the speakers my only option at this point?

Without knowing the situation...

You could buy a 4channel amp and then run the HU to that and that to the speaks. It'd be easiest to mount the NEW amp close to where the old one was.

ALL speakers have a + and - comming from some sort of amplifier. If you don't know how to tell the difference, use a battery and touch one wire to one side and the other wire to the other side of the battery...I'd use a little AA or AAA.