Sumer is the first "civilization."
Archaeologists have used many definitions of "civilization" over the years, but today most use it to mean urbanized, state-level societies.
A "civilization" has all of the following: (1) cities, with large, complex social organizations, themselves in much larger territories than in earlier times; and (2) centralized economies based on tribute and taxation; and (3) advances toward formal record-keeping, as well as some form of written script or a close alternative; and (4) impressive public buildings and monumental architecture, such as palaces and temples; and (5) some form of all-embracing state religion and ideology in which the ruler played a leading role, often as a living god.
By this definition, the first civilizations were Sumer and Egypt. People were farming in northern Iraq by 8,000 B.C.E., and the city of Eridu had a mud brick temple by 4,700 B.C.E. As early as 3,500, a secular ruler governed the city of Uruk, and writing evolved around 3,400. True Sumerian civilization evolved around 3,100 B.C.E.
Likewise, people farmed along the Nile River by 5,000 B.C.E., and Upper Egypt (in the south) consisted of several pre-dynastic kingdoms by 3,500 B.C.E. The first pharaoh of unified Egypt ruled in about 3,000 B.C.E.
TL;DR: there was no "civilization" before Sumer. Sumer became a civilization in approximately 3,100 B.C.E. Egypt evolved independently as a civilization around the same time.
What about humans before Sumer?
Modern humans (homo sapiens sapiens) evolved before 100,000 years ago. They lived as roving bands of hunter-gatherers until they learned to farm in approximately 10,000 B.C.E.
The inhabitants of the Abu Hureyra mound in Syria hunted gazelle to the exclusion of almost any other game. They also lived off about half a dozen staple grasses but knew of at least 200 others for eating, medical, and ritual purposes. To a considerable extent, they were managing and tending their environment well before they domesticated plants. Abu Hureyra began as a small village around 10,500 B.C.E. and flourished until 6,500 B.C.E.
Similarly, Jericho in the West Bank (Israel/Palestine) is indeed one of the most important antiquity sites that teaches us about the change to settled agriculture.
Underneath extensive Iron Age and Bronze Age occupation layers, Kenyon unearthed two much earlier farming settlements with substantial houses. The earliest of these, dating to earlier than 9,000 B.C.E., was a cluster of beehive-shaped huts. The closely packed dwellings nested inside a rock-cut ditch nearly 9 feet deep and a well-built stone wall complete with a tower.
The fortifications of this large village, which covered nearly 10 acres, are a mystery. They were either built for defense—Jericho lies astride a strategic route from the coast to the desert⎯or, perhaps, as flood control works.
After farming developed in Southwest Asia, it spread to Europe by 7,000 B.C.E. We know less about farming in East Asia, but archaeologists believe wild rice collectors in the Yangtze Valley turned to deliberate cultivation between 7,200 and 5,500 B.C.E. Archaeologists believe Mesoamericans domesticated squash by 8,000 B.C.E. People may have domesticated maize in the Mexican lowlands by 5,000 B.C.E. The Andeans tamed quinoa and the potato earlier than 2,000 B.C.E.
Okay, but what about art, culture, religion, and burial practices -- the other elements of proto-civilization?
Sometime between 40,000 and 30,000 years ago, Europeans first began making ornaments, such as beads, pendants, and perforated animal teeth to adorn their persons. At about the same time, the Cro-Magnons (modern European humans) started painting images of animals, signs, and anthropomorphic figures on cave walls.
The Cro-Magnons were brilliant artists in stone, antler, bone, clay, ivory, and probably wood. The Cro-Magnons are most famous for their rock art—paintings and engravings deep in the caves of southern France and Spain.
Humans had developed ideas of kin ties and the supernatural by 18,000 years ago. During the summers, neighboring groups came together at specific locations where game or salmon were abundant for a few weeks a year. At these times, groups arranged marriages, performed initiation ceremonies, practiced highly intense rituals, and invoked the ancestors and forces of the spiritual world to ensure the continuity of life. Shamans told tales and wove spells.
TL;DR: I know you didn't ask about Cro-Magnons and the Neolithic Revolution (invention of farming), but these are crucially important for understanding human technology and culture before the first civilizations arose in Sumer and Egypt.
Paleolithic and mesolithic humans (hunter-gatherers before humans invented farming) made beads, jewelry, decorated weapons, and magnificent rock art starting 40,000 years ago. They developed kin groups, marriage, story-telling, ceremonies, and shamanic proto-religions.
The oldest archaeological evidence of farming dates to Abu Hureyra in Syria in 10,000 B.C.E. People farmed near Jericho in the West Bank around 9,000 B.C.E. Farming developed in Europe, East Asia, and the Americas after that.
Human groups lived in tribes, settlements, and villages with this primitive art, social organization, and farming until Sumerian and Egyptian civilizations arose around 3,000 B.C.E.
Sources:
Brian Fagan, People of the Earth.
Bruce Smith, The Emergence of Agriculture.
Clive Gamble, The Palaeolithic Societies of Europe.
Kathleen Kenyon, Digging up Jericho.
Brian Fagan and Kenneth Garrett, Egypt of the Pharaohs.
Chris Scarre and Brian Fagan, Ancient Civilizations.
Paul Kwiwaczek, Babylon: Mesopotamia and the Birth of Civilization.
Samuel Kramer, The Sumerians.
Note to self: don't type about archaeology on my phone. Corrected multiple misspellings of the words "arch(a)eology" and "pharaoh."