Coins older than a 1,000 years can be discovered in many places of the world. The first coins were developed independently in Iron Age Antolia and Archaic Greece, India and China around the 7th and 6th centuries BCE. Coins spread rapidly in the 6th and 5th centuries BCE, throughout Greece and Persia, and further to the Balkans. (Wikipedia)
Coins were an evolution of “currency” systems of the Late Bronze Age, where standard-sized ingots, and tokens such as knife money, were used to store and transfer value. In the late Chinese Bronze Age, standardized cast tokens were made, such as those discovered in a tomb near Anyang. These were replicas in bronze of earlier Chinese currency, cowrie shells, so they were named Bronze Shell. (Wikipedia)
Standardized Roman currency was used throughout the Roman Empire. Important Roman gold and silver coins were continued into the Middle Ages (e.g. Gold dinar, Solidus, Aureus, Denarius). Anicent and early medieval coins in theory had the value of their metal content, although there have been many instances throughout history of the metal content of coins being debased, so that the inferior coins were worth less in metal than their face value. (Wikipedia)
You can get kits with all the tools you need to search for your own ancient coins and participate in an archaelogical dig.
Here are links for some terrific resources for Kids wanting to learn about ancient coins.