My research always starts with books. Usually ancient Greek or Roman texts in translation. Or texts from the Babylonian cuneiform tablets. I start by reading and reading and reading. Starting to see where there are overlaps: Where an icon from one culture appears in another, or where a similar story thread is echoed in another. It's a bit like tracing the roots of language (a language of mythology)
I have this view that in the modern Western world we treat ancient myths and stories like we treat the stories in texts like the Bible. We say "this probably didn't happen exactly as written, but we believe that these people were based on real people".
I now believe that this isn't true of the ancient myths.
I think that most of the oldest myths actually tell completely fictional stories that are drawn from celestial events (at least at their very beginning)
Is it likely that a man called Hercules existed and performed a series of tasks that made him renowned all over the ancient Mediterranean? Maybe... Until you realise there's a similar story that predates Herakles by about 800 years based on ancient Akkadian god myths. Then you find out that in this other myth the God is question is the Sun God, who travels around the globe and undertakes 13 tasks (one for each constellation of the Mesopotamian Zodiac)
Suddenly it seems much more likely that this is a metaphorical story that explains early celestial patterns.
Anyway, that's a small insight into the beginnings of the research process. After that i start by writing out the bare bones of the myths across all of the cultures (where parts are identical, where there is a different, special flavour) Finally i work out what story i want to tell. Trying to iron out the problematic parts of ancient stories that i think do not deserve to be carried forward (mostly the rape) Finding more imaginative ways around sexual violence.
Once i have a basic story thread i start to think about how it may work in performance. But for now i'll hold off on the full explanation of that area...