You may also want to post to r/AskAnthropology as the pre-historic movement of peoples and cultures is more of an anthropological question than a historic one. There's plenty of wiggle room and historians certainly deal with things early than this, but generally "history" starts when cultures start writing things down or appearing in written records, so the earliest you can really go with that is around 3000 BCE.

I'll also say that if you want Greece and Rome, 1200 BCE is going to be much to early for a cut off date. The traditional date founding of Rome is 753 BCE, and the earliest finds on the site of the city go back about 200 years before that. Ancient Greece, at least as most people think of it, begins with the reintroduction of writing to the Ionian Greeks around 800 BCE, the most famous examples coming from Homer's Illiad and Odyssey and Hesiod's Works and Days and Theogony. If you go back to 1200 BCE, you will find that there were Greeks, but at that time they are usually called Mycenaeans (after the greatest Greek city in the Illiad). They spoke an early form of the same language, but had a different system of writing and noticeably different culture than the later Classical period. Most of Ancient Egypt would fit in to that time frame though.

Now some suggestions. Like I said before, "history" doesn't often go back quite that early, so most of these suggestions will start around 5000 BCE at the earliest and move forward from there. It will also mostly be a list for Europe and the Near East, because you asked about those in particular and those are the parts of the world I study.

Books: A History of Ancient Egypt by Marc Va De Mieroop is a very good survey history, and A History of the Near East by the same author is also very good for covering the earliest civilizations with a written record in Sumer and Akkad, and the progression from there into Babylon and Assyria. As Sumer is the source of the first written records in history and you seem interested in origins, I recommend Sumer and the Sumerians by Harriet Crawford, and maybe Mesopotamia: Inventing the City by Gwendolyn Lick if you're interested in early urban development.

For Greece, I suggest The Aegean Bronze Age by Oliver Dickinson to get yourself into the Mycenaeans, A History of the Archaic Greek World by Jonathan Hall for early "Greek* history and Creators, Conquerors, and Citizens by Robin Waterfield for an easy read on all of Greek history before the Roman Period. For Rome, The Beginnings of Rome by TJ Cornell is the only book I've ever read dedicated to the period you asked about.

I'd also recommend The Horse, the Wheel, and Language by David Anthony to cover the expansion of the Indo-European language group as a good book covering the movements of people and cultures before written history.

Podcasts: Dominic Perry's The History of Egypt and Ryan Stitt's The History of Ancient Greece are both great at their subjects and both are hosted by academics with MAs in their subjects. They cover a lot more than the origins, but the early episodes of either show cover the early developments and they've done a couple of crossovers with each other discussing their shared history iirc.

I also strongly recommend Kevin Stroud's The History of English which does an excellent job of tracing the linguistic development of the English language and the people speaking it, starting with Proto-Indo-European in 4000 BCE.