Start with Karen Radner's Ancient Assyria: A Very Short Introduction and her free online course Organising an Empire: The Assyrian Way. Most of The Might that was Assyria by H.W.F. Saggs is still an excellent read, but be warned that the historical overview is now rather dated.

A Companion to Assyria is a very comprehensive overview of Assyrian history and society of all time periods, but it is a very dry read.

Other good general books on the Assyrians:

• Assyria to Iberia at the Dawn of the Classical Age, a Metropolitan Museum exhibition catalogue

• I Am Ashurbanipal: King of the World, King of Assyria, a British Museum exhibition catalogue

• The Campaigns of Sargon II, King of Assyria, 721–705 B.C. by Sarah Melville

• Sargon II, King of Assyria by Josette Elayi

• Sennacherib, King of Assyria by Josette Elayi

The major Assyrian capitals that have been excavated are Aššur, Khorsabad, Nineveh, and Nimrud. For information on Aššur, see Assyrian Origins: Discoveries at Ashur on the Tigris (free PDF). Nimrud: An Assyrian Imperial City Revealed by Joan and David Oates is the best available synopsis of the excavations at Nimrud. The recently published Nineveh, the Great City: Symbol of Beauty and Power provides a complete overview of Nineveh from the prehistoric to Parthian periods.

Additionally, there's chapters on Aššur and Nineveh in Mesopotamia: The Invention of the City by Gwendolyn Leick, and Royal Cities of the Biblical World has a pretty good chapter on Nineveh as well.

I'm happy to provide more suggestions if you have a particular interest in a topic like Assyrian religion, mathematics, art and architecture, military organization, literature, etc.