I have a Flash movie set to play once when my home page loads. (I made a 'play
again' button on the last frame of my movie if visitors want to play it again.)
Is there a way to set the movie NOT to play again if a visitor leaves the home
page and then comes back to the page again within the same user session?
Basically, I want to movie to autoplay only the first time they come to the
home page and not every time they reload/refresh the home page.

Help! I'm a newbie to Flash but have muddled through to this point.

What you could do is used a shared object. A shared object is a file written to
the local pc that contain various types of information.

You could design your flash file so that when it first loads it creates the
shared object and sets a variable to indicate that the movie has been played.

On consequent visits to your page, the flash file first looks for the shared
object and, having discovered it, reads the variable that indicates it should
not play

Look up some tutorials on shared objects to esplan how to set them up and use
them