Ancient Civilizations by Chris Scarre and Brian Fagan is a pretty good overview of the best attested ancient societies. I also highly recommend the superb exhibition catalogues produced by the Met Museum, both of which are available for free:

• Art of the First Cities: The Third Millennium B.C. from the Mediterranean to the Indus

• Beyond Babylon: Art, Trade, and Diplomacy in the Second Millennium B.C.

Van de Mieroop's A History of the Ancient Near East, ca. 3000-323 BC is the best place to start for the ancient Near East. I also recommend The Ancient Near East: History, Society and Economy by Mario Liverani. For Assyria and Babylonia specifically, see Ancient Assyria: A Very Short Introduction and A Short History of Babylon, both by Karen Radner.

Van de Mieroop's A History of Ancient Egypt and Wilkinson's The Rise and Fall of Ancient Egypt are the usual books used in Egyptian history courses. As a first introduction to Egypt, however, I recommend Temples, Tombs, and Hieroglyphs and Red Land, Black Land by Barbara Mertz. Also check out the excellent History of Egypt podcast, though it is currently only up to the 18th Dynasty.

For ancient Anatolia, I recommend combining Trevor Bryce's Warriors of Anatolia: A Concise History of the Hittites with Ancient Turkey by Antonio Sagona and Paul Zimansky. All of Bryce's books on ancient Anatolia are excellent, but he has always relied much more heavily on texts than archaeology, so the more archaeologically inclined volume by Sagona and Zimansky complements his history of the Hittites very nicely.

Edith Hall's Introducing the Ancient Greeks is a very engaging overview of ancient Greece that manages to incorporate a surprising amount of information without getting too bogged down in details.

See the AskHistorians reading list for more suggestions.