The Prologue explains that the play is set in Verona, Italy. In fact, you might think of the Prologue as a "cheat sheet" for the whole play. It tells you the setting and the most important parts of the plot:
Two households, both alike in dignity,
In fair Verona, where we lay our scene,
From ancient grudge break to new mutiny,
Where civil blood makes civil hands unclean.
From forth the fatal loins of these two foes
A pair of star-cross'd lovers take their life;
Whole misadventured piteous overthrows
Do with their death bury their parents' strife.
The fearful passage of their death-mark'd love,
And the continuance of their parents' rage,
Which, but their children's end, nought could remove,
Is now the two hours' traffic of our stage;
The which if you with patient ears attend,
What here shall miss, our toil shall strive to mend.
Some of the action also takes place in Mantua, another city in Northern Italy. Shakespeare doesn't tell us what year the play takes place, but based on the names and governments of the Italian city-states, it is sometime in Renaissance Italy -- fourteenth or fifteenth century (1300s-1400s).