If Mycenae is within your target range, John Chadwick's The Mycenaean World is a good place to start.
On the Homeric poems, to paraphrase the standard quip about Propertius, there are as many Homers as there are scholars. You should approach anything you read with extreme caution, because it's all speculation based mostly on the absence of evidence. But if you want to experience two wildly divergent views at the extremes of the continuum, you can do no better than to read M.L. West, The Making of the Iliad and Gregory Nagy, The Best of the Achaeans. For an antidote, you might try Barbara Graziosi, Inventing Homer, which is about the emergence of the idea of Homer in the late archaic and classical periods and beyond. A balanced book on the oral nature of the Homeric poems is Foley, Homer's Traditional Art. I highly recommend Jasper Griffin's Homer on Life and Death, which deals with the ideas underlying the Iliad (in particular), and doesn't attempt to sort out the mysterious origins of the poems. Troy and Homer, by Latacz and Bryce, Trojans and Their Neighbors are good introductions to the question of the historicity of the Trojan War, offering divergent views, with brief discussions of the origin of the Iliad. Bryce is more cautious than Latacz.
Loebs: Campbell's Greek Lyric series is very good. You'll want at least vol. I, Sappho and Alcaeus, vol. II, which includes Anacreon and Alcman, and vol. III, which includes Steisichorus, Ibycus and Simonides. The other volumes are post-archaic. You'll also want Gerber's Loeb Greek Elegaic Poetry alongside his Iambic. And Most's Hesiod is a must. Also West's Homeric Hymns. Whatever you do, don't skimp by buying used copies of the older Loeb lyric and iambic volumes, especially those edited by Edmonds. They're awful.
A selection of the more important fragments of archaic Greek poetry of all genres (including lyric, iambic and elegy) except epic, with more extensive notes than the Loebs but no translations, can be found in Campbell, Greek Lyric Poetry. Gerber's Companion to the Greek Lyric Poets is useful but very expensive, even in paperback.