The "rise of civilization" is not a single event, but a process of technological and cultural development that occurred over centuries. It's impossible to pin down a particular moment when any civilization was founded; they established themselves gradually over time. Often we date the founding of a civilization at the earliest writing; this is as much for convenience as anything else, as written records provide us with much richer details than we would get from studying other artifacts. It does create a substantial bias, however, because, by that definition, cultures such as the Navajo that never had a written language are not "civilized," even though the Navajo have many of the other features (such as agriculture, pottery, labor specialization, and government) we ordinarily associate with civilization.
The most important precursor for civilization is agriculture, which was invented in what is often called the Neolithic Revolution. Agriculture allows humans to maintain a large and steady supply of food in one particular place, and thus to grow much larger populations. Larger populations require more structure — government — and more opportunities for specialization, which creates a virtuous cycle of improved economic efficiency toward further output, more population growth, and still more specialization. In one form or another, this process has continued for thousands of years.
Still, it is notable that writing and other features arose around the same time in many places; it's still unclear how much contact there was between cultures of the Middle East, Africa, and Asia, so it is possible that these were in fact all offshoots of one still-more-ancient culture. Most scholars, however, think these civilizations were founded independently when and where conditions were appropriate. What we do know is that Latin American civilization was almost certainly founded independently; there's no plausible way people of different Latin American cultures could have been in contact during that time.
Early civilizations have a lot in common with one another, but also some important differences. Religion was very important and often directly tied to government, but the details of each religion vary tremendously. They tend to be founded along rivers or in river valleys, where fresh water is plentiful. They usually have some sort of founding text — a book or code of laws that sets down their religion and system of government. Literacy was exceedingly rare in this time (writing itself was a very new invention), and those who had it therefore had abilities most people couldn't understand. Thus, books appear to have been believed by many people to hold literal magic powers; we continue to see remnants of this thinking today in all sorts of tropes about "magic books" and "sorcerer's scrolls."
Another interesting similarity between Egyptian and Latin American civilizations that is often commented on is pyramids. Many people imagine some strange, supernatural or extraterrestrial explanation for this similarity, but the truth is much simpler: Pyramids are a basic and very stable geometric shape. They are easier to build than most other shapes, and more likely to remain standing over long periods of time. In fact, most modern skyscrapers are actually very steep, truncated pyramids which are slightly wider at the bottom than the top, rather than actually being true rectangular prisms of constant width.
Civilizations always appear to have been founded in particularly fertile regions (we don't think of the Middle East as very fertile today because its land has been drained over centuries, but thousands of years ago it had some of the most fertile land in the world), and often during times of historically unusual fertility. It may be that the abundance of food in such regions was necessary to take the risk and start the process of establishing permanent agriculture. Metalworking was also important, so accessible bronze (as bronze is the hardest metal that's easy to work) for making tools and weapons was another important factor. A variety of domesticated animals appears in almost every civilization; they perform work and provide food. The ability to find animals to domesticate (a wide variety of animals in the Middle East, llamas and alpacas in Latin America) was therefore an important factor in founding civilizations.
Ultimately, the most important factor may simply have been having someone in the tribe smart enough to think of the idea, and a tribe supportive enough of new ideas to listen. Every technology humans have ever invented ultimately began as somebody's crazy idea, and there's no reason to think bronzeworking, pottery, writing, or even agriculture were any different. It could have been a group who thought of it (simultaneous discovery is common even in science today), but it couldn't have been everyone. We may ultimately owe all of civilization to somebody who just had a spark of genius and realized you could plant seeds from plants and they would grow again, rather than just eating all the plants you have and not doing anything with the seeds.