The transition point started around the 1850s and early 1860s. In general you have several points along the transition: metal plating as armor over wooden hulls then fully metal hulls, steam power as one mode of propulsion then as the only method, and switching to turreted large bore guns ("artillery" as we've come to know it vs. simple cannons).

The use of armor on naval vessels dates back hundreds of years. We're somewhat biased towards the extremes of modern hardened steel armor as the only kind but using wood as armor was effective for considerable periods. Indeed, the USS Constitution was built with thick sections of white oak (a dense, rot resistant, and durable hardwood) and there are examples of cannon balls bouncing off of its hull in battle (hence the name "Old Ironsides"). And Korean "turtle ships", with a top covering of metal plates and spikes to defend against bombardment and boarding, have existed since at least the 1400s.

However, it was not until the French Gloire (1859) followed shortly by HMS Warrior (1860) that thick iron armor as hull protection began in earnest. There was only a brief window of a few years where armor plating over a wooden hull was feasible on warships before it was completely obsoleted by fully metal hulled vessels. This followed on the earlier innovations of adding steam power, especially via screw drive, to naval vessels.

Despite the UK and France pioneering naval modernization in their arms race (kicked off by the launching of the French vessel Napoleon in 1850 with steam powered screw drive in addition to its sails) it was the US Civil War which saw the first major battles between more modern vessels. At the Battle of Hampton Roads in 1862 the CSS Virginia (a "casemate" ironclad built on top of a wooden hull with fixed side mounted guns) faced off against Union vessels including the USS Monitor (with a fully iron hull and turreted guns). The battle, where multiple wooden Union ships were damaged or sunk and none of them were able to significantly damage the CSS Virginia, was pretty definitive proof that the only match for a steam powered ironclad warship was another steam powered ironclad warship.

Even though the Union/Confederacy/USA rapidly built up a more modern ironclad fleet through the 1860s they were still hampered by designs which were only well suited to coastal operations and weren't very good as blue water naval vessels. The UK and France were rapidly pushing ahead with vessels like HMS Devastation (a fully steam powered, iron hulled, ocean going capital ship with turreted big guns, launched in 1871) and the Redoutable (a fully steel hulled ship though still making use of a sail rig as a secondary source of propulsion, launched in 1876). The US would not catch up with proper open ocean cruisers and battleships until the Amphitrite-class in the 1890s.

In broad strokes you have a couple different phases of this evolution:

• 1850s: The first naval vessels employing industrial-era innovations, widespread (though not yet universal) appreciation that this was the way of the future, the stirrings of massive naval arms races between major naval powers.
• 1860s: The first major battles involving industrial-era naval vessels, the first build-outs of such naval vessels as cornerstone of naval fleets.
• 1870s: The first "modern" capital ships built and put into service (ocean going, sail-less, turreted large guns), a rapid transition of most major naval powers to these classes of ships, hitting stride on a worldwide arms race (involving the UK, France, Italy, Russia, and many others).
• 1880s: Continued escalation of the arms race, the first pre-dreadnought battleships coming into service.
• 1890s: The first major battles involving fully modernized battleships in the First Sino-Japanese War (1894) and the Spanish-American War (1898), providing very clear demonstrations of the devastating effectiveness of that modernization.
• 1900s: The naval modernization arms race kicks into high gear with the advent of "dreadnoughts" (heavy displacement, steam turbine driven battleships with all big gun main batteries). This, of course, carried through multiple iterations of rapid advancement obsoleting previous generations of ships on a very fast cadence up through WWI and carrying into the Washington Naval Treaty of 1922.