Read the sacred histories, Samuel and Kings, so that you are somewhat familiar with the flow of the Israelite timeline. Don't neglect the return from exile, Ezra & Nehemiah.
It isn't always important to know when a prophet wrote. Obadiah and Joel are not dated, and therefore we should conclude that it is not vital that we know their dates. You can create something of a timeline for the other prophets, here's a short list in order (without dates):
Obadiah
Joel
Jonah
Amos
Hosea
Isaiah
Micah
Nahum
Habakkuk
Zephaniah
Jeremiah
Lamentations
Daniel
Ezekiel
Haggai
Zechariah
Malachi
One thing to remember is that reading the prophets is usually like reading sermons (Daniel is a noteworthy exception; the Jews didn't even include his book among the prophets, but in the Writings). Read the books and portions of the books as occasional messages. The audience might be one group, or several different settings and assemblies. Isaiah has words of doom for the frivolous who only come to hear a popular preacher; his comfort is for those who weep for the sins of the people, and grow weary waiting for reformation and deliverance. Some of the prophets speak in particular against foreigners: Nahum, Obadiah, Jonah--but even so, the underlying message is to the covenant-people, either to comfort or to warn those who have ears to hear the Word of God.
Amos comes to town to confront the idle complacency of the northern tribes. Hosea is a long-lived witness to the slow-spiraling collapse of Israel-north. His book has (I believe) three parts: the living parable portion, Hosea, Gomer, and the children; a covenant-lawsuit at the center, in which the people's thorough apostasy and wholesale breaking of all of the covenant is laid out; and a lament for Ephraim.
The prophets to Judah cover the period before the exile, during, and after. The consistent message is a call to repentance, to cleave to God, and trust in him; wickedness brings wrath, righteousness (by forgiveness) is a way of life. Jeremiah is something of a counterpart to Hosea in terms of what he lived through in the south: the decline and fall of the house of David. Isaiah was more of a contemporary to Hosea. Ezekiel is a prophet to the exiles, most of whose hearts are not converted despite the trial. Haggai, Zechariah, & Malachi are all post-exilic, and continue the effort to reform a still-rebellious people, who are hard to make-ready for Messiah's coming.
Finally, I echo the principle of reading with a Messianic-consciousness. The hope of God's people is always in what the Promised One will do for them. And his salvation is dyed in moral or ethical color. This is often misunderstood as, "be good, so Messiah will come." This is how the Jewish faith down to this day reads the prophets. But the reality was (and is, with reference to the Second Coming), "Repent, no matter how good you think you are, because Messiah is coming."
