What was the script of Mesopotamia? What was its legacy and usage?

The principal languages of ancient Mesopotamia were Sumerian, Akkadian (Babylonian + Assyrian), Amorite, and - later - Aramaic.  They have come down to us in the "cuneiform" (i.e. wedge-shaped) script, deciphered by Henry Rawlinson and other scholars in the 1850s.  The subject which studies Mesopotamian languages and the sources written in them is called Assyriology.

Mesopotamian languages in the cuneiform script are mostly written on clay tablets, though they could also be carved on stone (example here).  Being incredibly durable, clay tablets have been recovered in thousands at archaeological sites from the Mediterranean to Bahrain to Iran.  More are found by the year.

As well as records of daily life and administration, they include religious, mathematical, musical and astronomical texts, the earliest known laws, and a rich literature that includes the Epic of Gilgamesh and the oldest versions of the Flood Story also known from the Bible.

As the world's first fully urban society, ancient Mesopotamia is of paramount interest to world archaeology, and its art, architecture and technology were the rival, and indeed often the precursors, of Egypt's. Mesopotamia was open on all sides to its neighbours, and its influence can be traced from India to Greece: the Pharaoh's scribes used cuneiform script to correspond with the Great Kings of the Hittites in Turkey, at Ugarit on the Syrian coast the forerunners of the Phoenicians kept their legal and commercial records on cuneiform tablets in Akkadian, and later the Biblical and Classical worlds grew up in the shadow of these ancient cultures to the east (and sometimes under their direct political domination).

Sumerian

Sumerian is an "agglutinating" language with no known relatives. It was spoken in South Iraq until it died out, probably around 2000 BC, giving way to Akkadian; but it survived as a scholarly and liturgical language, much like mediaeval Latin, until the very end of cuneiform in the late 1st millennium BC.

In the absence of related languages, Sumerian has had to be learned through the filter of Babylonian and Assyrian.  There are still many disagreements about what words mean, and how the verb behaves, but our knowledge of it is growing by the year.  There is  still no full dictionary of Sumerian, though the Sumerian-French lexicon recently posted online by the Swiss scholar Pascal Attinger is very useful.

There is no learner's grammar of Sumerian that can straightforwardly be recommended.  Non-specialists may find the excellent grammar of third-millennium BC Sumerian by the Dutch scholar Bram Jagersma heavy-going