Many historians go so far as to equate the term "Civilization" with writing. So let's look at that.
Egypt and Sumer (in Modern-day Iraq between the Tigris and Euphrates rivers) both founded literate civilizations around 3000 BC. There appears to be an ongoing debate over which was first. Its tough to know for sure, just because reconciling ancient civilizations' ruler-based chronologies with modern calendars (or worse yet, each other's chronologies) is a difficult challenge that occupies many a career.
When I was coming up most Historians said Sumer was first, but before I was born most Historians assumed it was Egypt. So it isn't hard to find older histories that state categorically it was one or the other. So I can see where you might get a confused picture from your reading.
The Indus Valley Civilization was probably 3rd, starting around 2600 BC. China started writing at roughly 1500 BC, and the Olmec Civilization in Central America also goes back to roughly that date. The first evidence of Andean Civilizations using their unique form of writing is from around 1000 BC.