For ancient Near Eastern history as a whole, I recommend A History of the Ancient Near East by Marc Van de Mieroop, The Ancient Near East: History, Society and Economy by Mario Liverani, and Civilizations of Ancient Iraq by Benjamin Foster.

For the Assyrians, begin with Ancient Assyria: A Very Short Introduction by Karen Radner and the brand new (as of last month) A Companion to Assyria edited by Eckart Frahm. Although it's rather dated at this point, I'm very fond of The Might That Was Assyria by Saggs. Finally, as a close look at a particular Neo-Assyrian king, Sarah Melville's The Campaigns of Sargon II is quite good.

For Babylonia, the best introductions for general readers are Babylonia: A Very Short Introduction by Trevor Bryce and Babylonians by H.W.F. Saggs (an updated and abridged version of his The Greatness That Was Babylon). Babylon by Joan Oates is a bit dry but also good. For the Old Babylonian period in particular, I recommend King Hammurabi of Babylon by Van de Mieroop and Mari and Karana by Stephanie Dalley.

Most Hittite scholarship is in German, but Trevor Bryce has produced some excellent works in English. The Kingdom of the Hittites is the standard history of the Hittites, and Life and Society in the Hittite World is a good overview of Hittite society. If you can find an affordable copy, Hittites: An Anatolian Empire edited by Metin Alparslan and Meltem Doğan-Alparslan is fantastic.

These should be plenty to get you started. I'd be happy to provide more reading recommendations if you have particular interests in art, religion, city life, medicine, literature, etc.