The Slinger is the first ranged unit in the game, and can be trained from the very start. However, it's not very effective - it can only attack adjacent tiles, and it is very weak on defense. The best way to use them is from defensible positions (Hills, Encampments, City Centers, etc.), because if any melee unit is allowed to attack the Slinger, it doesn't stand a chance. Also, it cannot stay behind your front line, because it doesn't have the range to shoot over them!
Fortunately, Slingers can be cheaply upgraded to Archers once Archery is researched. For this reason, you should establish a habit of training at least 3 Slingers every game, despite their ephemeral militaristic relevance. A Slinger costs 35 Production (equivalent to 140 Gold) and an Archer costs 60 Production (240 Gold), but it takes only 60 Gold to upgrade a Slinger into an Archer, which is an extremely good deal. So why at least 3 Slingers? Not only does it increase the chance you will get the Eureka for Archery, but when you upgrade those Slingers into Archers, you will get the Eureka for Machinery. This technology is crucial in every game, since it unlocks the Crossbowman (a premier military unit in the Medieval Era) and Kilwa Kisiwani, which is one of the best wonders in the game. Unlocking Machinery a few turns earlier than your opponents can mean a lot, and the Slinger plays an important, though not always obvious, role in it.
Killing a unit with a Slinger triggers the Eureka for Archery.
Like most early weapon systems, the sling evolved from a tool for hunting game to one for killing people. Simply a pouch between two lengths of cord, the sling was used to swing a projectile – a stone, lump of lead, or anything else convenient – in an arc until such time as it was launched. Although a thrown stone doesn’t have the lethality of an arrow or bullet, it can still do a lot of damage. If nothing else, a rain of well-aimed stones gives an enemy pause before rushing in. And, until composite bows were invented, it had a considerable range advantage over bows and spears. In addition, wasn’t hard to find ammunition. With the exception of Australia, ancient slings have been found in archaeological digs all over the world. Even a barbarian can sling a rock. And, as the First Book of Samuel points out, a slinger can bring down even a Goliath...