The AskHistorians reading list contains recommendations for the Sumerians, Babylonians, Hittites, Elamites, and Egyptians. I recommend beginning with the free Met Museum catalogues Art of the First Cities and Beyond Babylon, which provide great overviews of the 3rd and 2nd millennium BCE, respectively, accompanied by beautiful color photographs.

I haven't gotten around to updating reading suggestions for the Levant, but these are a good place to start:

• Canaanites by Jonathan Tubb

• Ancient Syria by Trevor Bryce

• The Archaeology of Syria: From Complex Hunter-Gatherers to Early Urban Societies by Peter Akkermans and Glenn Schwartz

• Early Cyprus: Crossroads of the Mediterranean by Vassos Karageorghis