Can anyone help be get readings off of my potentiometer with a push of a button?
I had a very similar issue an it is that you aren't setting up your button as an INPUT_PULLUP. When you press the button, you are sending 5V or ground through it depending on your circuit. But then it won't reset on release. That is why either you can put a physical resistor as the schematic shows here:

or you can simply change the button initialization to INPUT_PULLUP.