The earliest signs of cannons in Europe are form 1326, when we get one illustration (pic) and one written account mentioning them. So 1320's.
As far as I can tell hand-held gun showed up somewhere before the mid century, maybe in the forties. In the latter parts of the century saltpetre farms start popping up, ending reliance on imports form Indian mines.
A simple serpentine lever to hold the match is probably form the end of the 14th century, ca 1400 we can find it with a spring to hold it back. The basic touch-hole remains common until the end of the 15th century.
Simple dry-mix "serpentine" gunpowder is replaced by "knollen" wet-mixed somewhere in the early 15th century or so.
Breech-loaded cannons using removable chamber-pieces (keep a few pre-loaded spares around, hint hint) may have turned up as early as the start of the 15th century, hand-held variants where around by the middle.
The Hussite wars, 1421-1434, are in many ways when firearms really show their usefulness for the first time.
"Proper" stocks, instead of a staff for example, are IIRC a late 15th century thing, and about there we see the proper matchlock appear as well. Taken together, somewhere close to the turn of the century, we get the arquebus.
Rifled barrels late 15th century, wheellock firearms are first mentioned in 1507, the pepperbox was around by 1530 if not earlier, revolvers (usually long guns, very rarely pistols) before the end of the 16th century. Snaphance lock somewhere mid-late 16th century IIRC, turning into the flintlock proper towards the end of the century(?).
The musket first turns up as a very large cousin to the arquebus in the 16th century, perhaps having started out as a wall gun brought into open battle. It then shrinks a bit and takes over as the main infantry gun from the arquebus, and then keeps shrinking until by the mid century or so it's in many ways the arquebus come again in function.