After a good bit of digging I figured out how to modify the 'edit-user.php' page in the admin area such that it now has the additional information I want for users and no longer shows information that I am not interested in, but I cannot seem to find any way to modify the 'user-new.php' page to match.
In the case of the edit-user.php changes, I was able to add functions that used the
add_action( 'show_user_profile', 'value' ) hook to add new items to the page, and a combination of unset() and some jQuery to remove some of the fields.
So far I have not been able to find anything similar that I could use to modify the 'user-new.php' page to achieve similar results. Does anyone know how this can be achieved outside of making changes to the core?