The earliest firearms used arrows, and later, stones as projectiles, as stated above. Lead becomes the most common projectile due to a number of reasons: its ease of use (one could melt down lead over a campfire without much smithing work, and if you already had a mold, you could make your own shot fairly easily), its relative hardness (lead was hard enough when cool to be able to pierce most animals and people for hunting/war/killing, with more powder needed for more power (to an extent)). Lead was also soft enough that it had the ability to bloom or mushroom upon impact, so kills were more likely to occur (although to what extent this wasn't an added bonus, I have not yet found. That doesn't mean the information isn't out there, and it was certainly known by the close of the 19th century). Lead was also far cheaper than steel, which was an added bonus, and heavier (meaning it retained energy longer) than any other metal in use for projectiles until the 20th century.

Steel's popularity for projectiles grew in the 19th century as powder technology grew, allowing for faster and faster propellants to be used. With a faster propellant, you could now use a lighter projectile, which was part of the theory behind the Minié ball (an early conical bullet with hollow base, which expanded to grip the rifling slightly more secure for a more accurate round.) Leading up to the mid 1800s, rifle rounds were staying stable at about .58 - .75 caliber, with pistol rounds being between .32 and .68 caliber, roughly. As powder quality increased, caliber decreased (although effect was still similar.) Now, a .44 Henry round in 1862 was just as effective as a .68 caliber Kentucky long rifle was. Lead was still traditionally used due to cost, availability, and malleability (due to variances in barrel technology, a .44 barrel could be between .435 and .455 inches across, land to land. Lead was needed to make that tight seal.) By about 1880, steel started appearing more and more commonly in small calibers, and steel's relative hardness compared to lead made it preferable as a penetrator projectile, even over the newer copper-jacketed lead common in military ammunition by 1903.

EDIT: there was also some experimentation with alternate forms of projectiles in the 1850s and again in the 1950s, both because of major technological gains in the form of powder technology. The 1850s saw the creation and quick decline of the Volcanic "Rocket Ball," made by the Volcanic Arms Company, which was quickly absorbed into Smith & Wesson. The rocket ball was a similar design to the Minié ball, but with a hollow core throughout the lead projectile. This core was filled with powder and a primer cap was placed on the end, sealing the cartridge. This was realistically the first self-contained metallic cartridge, but was quickly abandoned as a design, mostly due to lack of "stopping power" - about half the strength of the modern .25acp, easily the weakest smokeless powder round available on market.

The second major change came with the Gyrojet pistol, a veritable "space gun" design centered entirely around a copper rocket. The rocket was pushed back by the hammer onto the fixed firing pin, which ignited the primer and pushed the rocket out. Because of how slow the projectile was, it could realistically be stopped in the barrel with firm enough pressure. Effective distance was about 75 yards, but it was really only "deadly" after 30 feet - again, because of slow ignition.

Finally, the last real technological branch brings the rocket ball back to life. In the 1980s, The German G11 was treated for adoption by the BRD, and it prominently featured caseless ammunition, where the (copper-jacketed lead, because of military use) projectile was encased entirely within the powder, with a primer embedded in the base of the bullet that stayed with the projectile. Due to a number of issues, among them overheating, overcomplication of the cartridge, and ignition issues, caseless ammunition was quickly shelved again for later approach with the failure of the program in 1990.

While on the subject of ignition systems (I'm not going to ramble too much), there was a very brief electronic primer trial, with early examples surfacing in 1994. The standard primer uses a mercury fulminate mixture combined with pressure to ignite, while the new electronic version used a steel primer and a firing pin connected to a battery. The firing pin could barely touch the primer, charge it, and ignite the powder a few thousandths of a second faster, and due to not filling a role needed by most anybody, was quickly abandoned as well, although a few thousand rounds did make it to civilian markets by 1998.

Early cannons used arrows, and then later stones. I haven't been able to find out much on whether handgonnes in particular used stone balls at any point, but in Sean MacLachlan's Medieval Handgonnes there's a reference to smaller cannons typically using lead shot and large cannons using stone shot, for reasons of economy and ease of manufacture.

So, in all likelihood, stone may have been used in handgonnes, but lead was probably the most common material for projectiles in small caliber guns.