I was wondering if there was a way for me to have one button (just a regular old pushbutton from a hobby shop) cycle through commands each time it is pressed.
Yes, there is.
Would I just use the ButtonStateChange as stated here?
Yes.
Or is there something else that is more efficient that I could use?
What could be more efficient that detecting, and counting, changes of state? I suspect you meant simpler, as in "Is there a function that will do all the work for me?". The answer to that is no.
I really need help with this as I am trying to get this project going and I have everything else working but this one button which needs to cycle through three commands.
What do you need help with? All that is missing is making the code do something when the value of buttonPushCounter is 0, 1, 2, or 3. Only you know what you want it to do.