I wrote about this in What’s the best modern translation of the Epic of Gilgamesh?

Sumerian history and archaeology:

• Sumer and the Sumerians by Harriet Crawford. This is currently the best single volume introduction to the Sumerians. Crawford examines topics like settlements, temples, trade, and manufacturing primarily from an archaeological perspective.

• Early Mesopotamia: Society and Economy at the Dawn of History by J. Nicholas Postgate. This is an excellent overview of Mesopotamia from the Uruk period to the end of the Old Babylonian period. Postgate expertly balances archaeological and textual sources and provides overviews of early Mesopotamian towns, palaces, temples, agriculture, crafts, warfare, and laws and political organization.

• Ancient Mesopotamia by Susan Pollock. This is a very dry read, but the book contains an excellent analysis of the growth and development of urban settlements in Mesopotamia as well as discussions of early Mesopotamian administration, economy, and ideology.

• The Sumerian World edited by Harriet Crawford, a collection of essays covering almost all aspects of Sumerian society.

For an overview of ancient Near Eastern history in general, I recommend picking up one of the following books.

• The Ancient Near East: History, Society and Economy by Mario Liverani

• A History of the Ancient Near East, ca. 3000-323 BC by Marc Van de Mieroop

• Ancient Near Eastern History and Culture by William Stiebing and Susan Helft

The Stiebing and Helft book is the most approachable and readable, Liverani's book is the most detailed of the three, and Van de Mieroop's book falls between the two (offering a fair amount of detail while avoiding getting too bogged down in minutiae).

Finally, a good free resource is Art of the First Cities: The Third Millennium B.C. from the Mediterranean to the Indus published by the Met Museum.