The thing that most caught my attention, though, was the milieu: the very early days of the Roman Republic, within a generation of the overthrow of the last king. Primary sources from this misty time are extremely sparse, and the story of Coriolanus is only known from accounts by historians writing centuries after it supposedly took place.
At the time of the play, Rome is still an infant polity; I don't know the real distances, but the feel in the play is that a couple of days' march from the Capitol takes you out of Roman territory. Italy is tiled with similar city-states; no contemporary observer could have selected Rome as the one that would expand to surround the entire Mediterranean.
The politics is distressing, with tension between the military aristocracy that (at least in principle) protects the city and countryside of Rome, and the mostly-unarmed people who do all the productive work. Patrician/plebeian interactions are weirdly reminiscent of parts of A Tale of Two Cities. Shakespeare's sympathies are hard to place: no one escapes criticism.
Now I have begun another less-well-known play, Cymbeline. This one is set in Celtic Britain a couple of generations before the Roman conquest.