Can anyone tell me the names of some Romantic, Pre-modern, and Modern plays that might be a research subject?
There's a pretty broad range here. I sugget "Death of a Salesman" or "A Doll's House" because I happen to think both are brilliant social commentary. "Salesman" scared me to death when I read it the first time.
I would look at "The Crucible" in regards to hysteria (very common in today's society), "A Doll's House" (regarding the woman's role in society and oppression), "Streamers" (homosexuality and mentality in the military). One last play, that I personally love, is "A Streetcar Named Desire" (deals with mental defect, deceit, and relationships).
A good pre-modern play would be Everyman, or perhaps The Second Shepherd's Play.
A Romantic play worth examing, although it is mainly a closet-drama is The Cenci, by Percy Bysshe Shelley. The same author's Prometheus Unbound is a poetic drama also worth reading.
Among modern plays, almost any of the major plays or Tennessee Williams or Eugene O'Neill would be a good investment of your time and energy.
Good luck with your project!
Hopefully you have already read a few plays that interest you in some way. I would start with those of which you are already familiar. Having a previous understanding of a play will certainly reduce your research and the multiple rereadings necessary for such an assignment.