I have a 3-wire 220V going to my well house. I am wanting to come off the 220 to a 110 outlet. I realize this is not the safest thing to do but it is only temporary.
My question is, I know I use one wire so I cut let's say the white wire and connect it to one side of this GFI outlet, Do I connect the other end of the white wire to the other side of the GFI to get the 110? The 3rd wire is a solid copper wire and does not appear to even be connected anywhere onto the well connections but I am not sure if that is a "ground wire" or a "neutral". Can anyone tell me how I do this so I don't screw things up please?

First it would appear you have no idea, so my best answer is don't do it.

220 wiring has three wires, two "hot" wires at 110 each and one ground wire. To get 110 you merely use one of the hot wires and the ground wire.

I will say this as to why do you want to do this, what are you going to run, is it going to run at the same time as the 220 well pump,
In general my best advice is to use a professional if you are not trained and have knowledge in what you are doing.

There is no neutral, the white wire you want to cut is 120 volts, the other is 120 volts , and across both is the 240 volts. The third wire probably is suppose to be the ground for the well motor or casing.

The GFI will not work properly.

This is not permitted. The GFI detects and trips on ground current and your scenerio involves creating 120V by using the ground for current.

A good alternative would be to use a portable generator or an extension cord.