Kramer's works are more than 50 years out of date. The theory of a "Proto-Euphratean" substrate in Sumerian has been mostly debunked and rejected, for example. Additionally, excavations in Syria and southern Turkey in recent decades have revealed that urbanism developed at least as early in northern Mesopotamia (e.g. Tell Brak) as in Sumer, and northern Mesopotamia was actually ahead of the south in some ways - early administration and the development of sealing practices, for example. Some of the recent scholarly developments are more subtle, like the royal name Urukagina now being read as Uruinimgina. Kramer was a superb Assyriologist, and his works are still well worth a read, but be sure to supplement them with the standard works on the Sumerians published in the last couple of decades.

I added some books on the Sumerians to the AskHistorians reading list a while back, so be sure to check that out. My suggestions:

• 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.

• Ancient Mesopotamia by Susan Pollock. This is a fairly dry read, but Pollock's book has 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. The volume contains a collection of essays covering almost all aspects of Sumerian society. The chapters are readable, concise, and written by experts on the topics. Each chapter is accompanied by a helpful bibliography.

• 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.

For an overview of ancient Near Eastern history in general, I recommend picking up one of the following books. The Stiebing and Helft book is probably the best book for those first learning about Mesopotamia, 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).

• 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

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.