My Nikon 8000s has an Auto/Manual focusing selector on the body
(actually, two modes of auto), as does my new Nikon D70s. Each of the
lenses for the D70s also have an A/M focusing selector.

I've read the instructions for both the camera and for the lenses, and
I'm still confused. Do I need to set both to A for auto focusing, and
both to M for manual focusing, or what?

What happens if the camera is set to M and the lens to A?

What happens if the camera is set to A and the lens to M?