The Sumerians are widely credited with being the first real civilisation on Earth, beginning in around the 5th millennium BC. Cities and agricultural communities existed before this time, but are generally not considered to have constituted a civilisation. The Sumerians, who were situated in modern-day lower Iraq and Kuwait, are widely believed to have invented the wheel, writing, government, law, systematic agriculture, and irrigation, among various other early innovations and marks of civilisation.
Although the dates of the start of these civilisations are generally not well-defined, the order of the appearance of "civilisations" is often accepted as the following:
Ancient Sumerians
Ancient Egyptians
Akkadians -- arguable since they largely took over everything from the Sumerians in the same region
Ancient Indians / Indus Valley Civilisation
Ancient Chinese
Ancient Greeks (Minoans and then Myceneans) -- arguably older than the above
Phoenicians/Caananites
Ancient Romans
Ancient Persians/Iranians