I am making a simple lamp I want to install a button switch on it. Right now it simply just turns on by plugging it in but I would like to have it plugged in at all times and have the button switch to turn it OFF and ON.

I would normally just guess on how to install it but I want to make sure I'm doing it correctly. The button switch has two wires coming out of it, both the same color.

Do I cut one of the wires on lamp and piece the button into it?

The lamp cord has a black and a white conductor or a smooth and ridged side. The black wire or smooth wire is the one to cut. Since you apparently are not proficient with wiring, be very careful and unplug the lamp first.

A feed through type cord switch as shown below, would be neater looking than any switch that has external wires.