Israel and Judah
The Book of Isaiah is set in Judah during the reigns of King Ahaz and King Hezekiah, around the eighth century BCE, and it's often accepted that this is the period when Proto-Isaiah (chapters 1-39) was composed. It's a period where Judah is seeing itself threatened by various outside powers: first, the northern kingdom of Israel's alliance with Ephraim and then the invading Assyrian forces of Sennacherib. The Babylonian invasion and exile is also just around the corner.
So, during this time period, people are struggling to hold on to their own identities and not give into all this outside pressure. Particularly, they're trying to remain faithful to their own God and avoid polytheism. You can see this come up repeatedly throughout the story. Ahaz has to ask God for a sign, turn his attention towards his own deity and away from other preoccupations. Hezekiah, too, is rewarded for (in the end) helping Judah bring itself back to its own sense of identity, its adherence to one God. And for his part, Isaiah is the man who continually calls people back to this obligation, reminding them of their true identity. The setting of Judah during this time period is in many ways the perfect backdrop to Isaiah's lessons about the importance of God to the lives of his readers.
Isaiah: The Sequel(s)
Now, that's just Proto-Isaiah. Other people would zoom in to take up the Isaiah mantle, and keep the prophecy coming.
Deutero-Isaiah and Trito-Isaiah (come out, come out, whoever you are) were probably writing at some point during or after the Babylonian exile, when things were really going belly-up. Now, Proto-Isaiah expressed the fervent and heartfelt belief that Jerusalem would never ever fall to its enemies. Funny you should say that, Proto-Isaiah... In 586 BCE, the Babylonian empire tromped in and razed Jerusalem to the ground, sending its citizens into exile in Babylon. Translation: enormous tragedy for Israel.
Deutero-Isaiah is a hopeful dude (or dudette—there were female prophets in Israel, albeit very few), who writes how the Persians will allow Israel to return home and rebuild their Temple, which does happen in 539 BCE. It's all very excited and hopeful, and this is where the Suffering Servant and a lot of the Messianic language pops up (tl;dr of Deutero Isaiah: It gets better.)
But, by chapter 54, Trito-Isaiah kicks in, and that smile turns upside down. Trito-Isaiah seems to be frustrated that the exile is over, and yet nothing else has really changed. He's still not sure what the point of the whole exile thing was, and why God hasn't stepped in on behalf of his peeps. (tl;dr of Trito-Isaiah: Was this really necessary?)
So, there are really three settings in the Book of Isaiah, but it's all the same rollercoaster. There are peaks, crazy loop-de-loops, and death-defying drops, but hey: that's Israelite history for you.