What Color Background Should I Use?

You need to adjust you camera setting( if your camera allows it). From what you have said I gather the camera is metering across the whole frame. Somewhere in the menus should be a setting that allows you to meter just the center of the frame.

{I use a mirror less camera so what I see in the view finder or on the back-screen is exactly what I get in the photo so I ignore the metering and just adjust the exposure until the coin looks right. If I have any doubt I turn on zebra. I am guessign your camera is a DSLR so you probably don't have those options}

I use a black/dark gray non-reflective background.
Why?
Because I want as much light as possible coming from the coin only.
A white background will tend to wash out the colors/toning of a coins surface.

I use a sheet of black kindergarten construction paper for my background.

Agree with pep... you want the light to be measured from the coin, not the background. So a manual exposure setting will give you the same result, whether the background you use is white, black, or somewhere in-between. Adjust from that if necessary, but that's the starting point.

If you're using manual settings and getting a change based on your background color, you're not really using manual. What you're doing is looking at the meter and adjusting your settings based on setting the needle to the middle of the meter. That's not manual, that's "manually setting the speed, aperture and ISO based on the light meter, just like the camera would if you let it". You might as well use an Auto mode if that's how you do it. Shooting manual means getting your exposure settings based on 18% gray or some other fixed mechanism, and then *ignoring* what the meter says because you know better than the meter does.

So yes, get that 18% card, but set your exposure from it once and don't use your meter for exposure. I will say, there's a large contingent of folks that would recommend automatic exposure settings rather than manual. That can be done too, I just don't prefer it.

that is also my approach, I use camera flock which reflects very little light.

I don't think I ever mentioned exposure compensation, but it's good to hear you're reading!