Oh, Something I feel confident contributing to!
[A bit nervous as this is my first post so mods please feel free to remove this if it doesn't live up to the Sub's standards]
The task of defining the first civilizations is difficult.
It seems simple enough, just ask "when were the first cities founded?".
Then you have to ask, how do you define a city? You might answer that a city is "a large group of people with division of Labor and agriculture."
But this definition too holds its own problems, because the agricultural revolution as a definable event or even series of events more than likely didn't occur as popular culture would lead one to believe.
The site at Çatalhöyük (cha-tal-hoy-yook) in the Konya province of southern Turkey is considered by many to be one of the oldest human cities or proto-cities, for which we have extensive archaeological evidence. It maintained a population of around 5,000-10,000 during its occupation from around 7500BCE to ~5500BCE and featured many of the hallmarks of what we would consider a civilization. The farmed, we have small but some evidence that they kept livestock, there is extensive evidence of religious practice and consistent funerary practices, as Neil Roberts and Arlene Rosen cite trade in obsidian across the Neolithic middle east in obsidian that was originated from the area surrounding Çatalhöyük. However, where this site seems to be the first fully fledged city archaeological evidence seems to suggest that the residents were neither fully agricultural nor fully hunter gatherers. Instead, the population underwent a process of breaking apart migrating and returning with the seasons.
Now, what this means in order to answer your question. If we look at the earliest forms of human settlement on a large scale, the process of transitioning from a Hunter gatherer society to an agricultural one was a slow process that took generations and they more often than not had contact with other peoples and even engaged in trade of natural resources. So they would not have seen themselves as the first to do any significant act they just slowly moved away from one form of subsistence until it was less practiced and eventually abandoned all together.
I hope that adequately answers your questions.
SOURCES: Roberts, Neil, and Arlene Rosen. "Diversity and Complexity in Early Farming Communities of Southwest Asia: New Insights into the Economic and Environmental Basis of Neolithic Çatalhöyük." Current Anthropology 50, no. 3 (June 2009): 393-402. doi:10.1086/598606.
Price, T. Douglas, and Gary M. Feinman. Images of the past. 5th ed. Mountain View, CA: Mayfield Pub., 2006.
EDIT: fixed my dates
As a followup, how sure even are we that those are the first civilizations?
So where would nomadic tribes and steppe people fit into that framework?
What makes the Middle East such a hot-spot for early civilizations? Doesn't northern Africa or south east Asia or even the America's have equally suitable living conditions?