Historical Firsts in Money
First representative money: The first form of representative currency seems to have arisen in Ancient Mesopotamia (c. 3000 BCE); individuals could bank assets (e.g.farmers could deposit grain) and receive a record of credit on clay tablets which could then be used to pay for something else.
First coin: Depending on how strict you are with your definition of coin, the first metal tokens used as money – the Chinese Bronze Shell/Tong bei (銅貝) from about 1100 BCE – could qualify. These were shaped like the cowry shells that had preceded them, however.
The first metal tokens to be used as money that were actually coin-shaped are from Iron Age Anatolia (late 7th century BCE). Historians can't be sure that these were actually used as currency, but it's probably safe to say that the first coins were produced and used sometime in the 600s BCE. The earliest inscribed coins are from Ephesus in Ionia and dated 625 – 600 BCE.
First paper money: The first documented use of paper money was in China during the 7th century (CE) onwards. This money seems to have been representative currency. (There is rumour that Carthage issued government notes on parchment or leather before 146 BC.)
First government-issued fiat banknote: The first ever government-issued paper banknotes (that count as fiat* currency) were used in China during the Yuan dynasty; the Zhizheng Jiaochao (至正交鈔) notes were first issued in 1350 CE.
Image source (photographed at Tokyo Currency Museum)
*Fiat money is currency that has only nominal value; it has no intrinsic value and is not legally exchangeable for something else with intrinsic value.
Sources: 'History of money' on Wikipedia, 'Coin' on Wikipedia, 'Jiaochao' on Wikipedia, 'Paper Money' on Wikipedia