The transition you describe was a long process. The biggest switch was from muzzle-loading (i.e. put the bullet in the far end of your gun) rifles with loose ammunition to breech-loading (load it at your end) rifles with cartridges. While people had been trying to make this work on a large scale for decades, it was the Dreyse needle-gun as adopted by Prussia in the late 1840s that brought it to broad attention. During the Austro-Prussian War in 1866, Prussian soldiers had a dramatically higher rate of fire than their Austrian enemies, and could load lying flat while the Austrians had to stand. The rest of Europe quickly followed suit

While a dramatic change, this wasn't quite as important as you might think, because artillery was seeing an even more significant increase in firepower. During the Franco-Prussian War, the French Chassepot rifles outranged the German needle guns, but the vastly superior German Krupp artillery meant massed infantry could be simply blown apart. Machine guns likewise would later mean masses of men were at serious risk. Infantry was forced to abandon massed formations on the battlefield, and cavalry became less and less relevant.

The rate of rifle fire actually didn't increase all that much for a long time afterwards. Single-shot rifles were replaced by those with five- or ten-round magazines, but bolt-action rifles (you have to do something manually between each shot) were universally used in WWI. The really interesting development came near the end of WWI, when the Germans started issuing submachine guns to their specialized assault troops. While firing a bullet completely inferior in range and accuracy to a rifle, they were capable of fully automatic fire (fires constantly while the trigger is held, like a machine gun.) This was a huge advantage in the kind of close-quarters combat seen in trench warfare, and submachine guns took on a role alongside rifles. (Although once again, it wasn't the most important thing happening. Germany still lost, partly because of another crucial innovation by France and Britain: Tanks.)

By WWII, interest in semi-automatic rifles (shoots once per trigger pull and that's it) was increasing. Most armies still relied on bolt-action rifles, but the U.S. switched to semi-auto in the 30s, and the Soviets were in the process of switching when the Germans invaded (and had to restart production of their bolt-action, to equip millions of new soldiers.) Meanwhile, Germany realized that a lot of engagements were taking places at ranges where a rifle was overkill, but a submachine gun wasn't quite enough. This led to the first assault rifle, a gun intermediate in range/power between a rifle and submachine gun, and capable of selective fire (switching between semi-automatic and fully-automatic fire.) Assault rifles subsequently became dominant, with the Soviet AK-47 and related models being made in staggering quantities.

TL;DR: Slowly, but with one big leap around the 1860s, then another in the 1940s.