I cannot, for the life of me, figure out how to format a post so that the thumbnail featured image shows.

Excerpts of the posts clearly display the featured image. The featured image is absent when the visitor is viewing the post.

I have searched high and low through your wonderful forum, but have not been able to locate a solutions.

Have looked up what a featured image is for on wordpress.org ?
Have you tried another theme like twenty thirteen to see what it does ?

Yes, I have reviewed what the featured image is, and how to insert it. This only gives me two featured images on the post excerpt display.

Yes, when I go to Twenty Twelve, the featured image appears on the post page.

Is there a way to have the featured image appear on the post page?

Atahualpa uses the featured image when you are displaying excerpts on a posts page.