As a preliminary you should read the commentary for Chapter 1. Pay particular attention to the division of myth into Myth Proper, Saga or Legend, and Folktale.
Investigate the offerings at the following site: http://www.gods-heros-myth.com/mythmore.html. Read through the general description of Greek and Roman Mythology and study the entries under Mythical Scholars to familiarize yourself with some of the foremost interpreters of myth.
Browse through the offerings at: http://www.stoa.org/diotima. It remains a very important site for the study of women and gender in the ancient world. Its purview includes every facet of antiquity: literature, art, history, religion, as well as mythology. It should be consulted should you wish to write a research paper on the role of women in ancient Greece or Rome.
Investigate Interpretations of Greek Mythology, edited by J. Bremmer, which contains a collection of essays that use various methodologies to approach the study of myth.
Read Roberto Calasso's The Marriage of Cadmus and Harmony. It is an interesting, rewarding, artistic meditation on the nature of myth, the way myth is shaped and reshaped, and the contemporary significance of myth for understanding our world.
Consult the first two chapters of Graf, F. Greek Mythology: An Introduction for a more complete understanding of myth and the scholarly approaches that have been brought to bear on this subject.
Read G. S. Kirk's The Nature of Greek Myths for a sober, critical account of the various approaches to myth, their strengths, as well as their limitations.
For a solid introduction on the role of women in Greek mythology consult Mary R. Lefkowitiz's Women in Greek Myth.
In Byrne Fone's Homophobia: A History read the chapter dealing with homophobia in Ancient Greece. Assess his interpretations of the evidence in light of Dover's Greek Homsexuality.
How would you judge Ivan Strenski's criticism of mythographers and mythography in his Four Theories of Myth in Twentieth-Century History?
Assess Cythia Eller's criticism of the theory imagining a gynocentric golden age before the onslaught of patriarchy, as expounded by Marija Gimbutas.
View either Joseph Campbell and the Power of Myth with Bill Moyers or The World of Joseph Campbell: Transformations of Myth Through Time. Assess his comparative approach to world mythologies. How important is Greek and Roman mythology for his archetypal theories?