Until the 1880's it really wasnt doing anything. The ships were mostly paid off, and all those fancy ironclads (which even by the end of the war were 2nd rate) were mostly rusting in harbor. European navies continued to innovate and expand with full steel ships, larger guns, advanced machinery and designs.

It wasnt until the Garfield presidency in the 1880's that the Navy began to receive the attention it deserved. The first attempt to make modern steel warships resulted in the ABCD ships the (Atlanta, Boston, Chicago, and Dolphin) 3 cruisers and a dispatch ship all made of steel with steam engines, but also mounting masts and sails. They were combined into the Squadron of Evolution (http://en.wikipedia.org/wiki/Squadron_of_Evolution#mediaviewer/File:Chicago._Leading_the_Squadron_of_Evolution,1889-NARA-_512952.tif)

However the USN still lacked any even 2nd rate battleships (in the quality sense). In fact Brazil's purchase of a brand new British built battleship mounting 9.2in guns made her the best equipped naval power in the hemisphere. SO 2 new battleships were ordered built, while not up to the standard of new European ships they were a step in the right direction, these would be the ill fated Maine, and her half sister Texas. Though a drawn out construction period meant they slipped further behind other navies, made worse by problems in getting sufficient industrial bases in the US for quality steel and guns.

Then in 1890 came the 3 ship Indiana class which while even more advanced was designed as a coastal warship and they all lacked sufficient free board and had ineffective armor scheme. The following single ship USS Iowa commissioned only in 1897 was the first true sea going battleship in US service.

Additionally the US had been constructing classes of both Armored and Protected cruisers. The difference being that Armored cruisers had a heavy belt of armor to protect as much as possible, while Protected cruisers concentrated it around machinery and weapons.

The only notable actions were in relation to the Spanish American War in 1898. In the waters around Cuba the USN concentrated its battleships such as they were (including the Oregon sailing 14k miles around South America from the west coast). And protected the landing o the expeditionary force. Then destroyed the poorly equipped and outclassed Spanish Squadron as it attempted to flee Santiago bay.

In the Philippines then Commodore Dewey led his Asiatic squadron to much the same result against the Spanish squadron in Manilla Bay. Dewey's flagship the USS Olympia is still afloat as a museum ship in Philadelphia and is the last Protected cruiser left and gives a good idea of the USN at the time.

Other notes would be that in 1890 Alfred Thayer Mahan published his "The Influence of Sea Power Upon History" which in many respects dictates naval policy to this day. And in 1907 Teddy Roosevelt ordered the battle fleet on an around the world cruise to demonstrate that they could after the Russians had encountered so much difficulty. The Great White Fleet was a great success and spread a deal of goodwill. But by the time they had returned HMS Dreadnought had already been launched and the USN was building their own response and every ship in the fleet was obsolete.

EDIT FOR SOURCES FROM FLAIR APPLICATION:

US Cruisers 1983-1904: Birth of the New Steel Navy by Lawrence Burr

Manila and Santiago: The New Steel Navy in the Spanish-American War by James Reckner

And The New American Navy by John D Long (SECNAV during the war)

A notable example of the US Navy was the Great White Fleet that was commissioned by Theodore Roosevelt and ordered to circumnavigate the globe (1907-1909) during his presidency.

Prior to this, the US Navy consisted of only 90 ships with almost a third of them wooden. Congress, hoping to enforce treaties, and protect overseas holdings among these were 16 steel battleships all painted in peacetime colours, which at the time happened to be bright white. Hence The Great White Fleet.

Ever since the Japanese defeated the Russians during the Battle of Tsushima in 1905, Japan had grown as a Naval power. Therefore, one of the purposes of the White Fleet was to show Japan that the US Navy could be deployed any time, anywhere, even from Atlantic ports, and they shouldn't get any ideas. And it worked. At least for the next 34 years.

While this was the largest American show of force at the time, it wasn't the only one. During the Algeciras Conference in 1906, Roosevelt ordered eight battleships to patrol the Mediterranean.

While the White Fleet was considered obsolescent by the time it returned home, the fleets trip resulted in massive changes to American Naval policy. Including a complete overhaul to Warship design, and long term deployment, which would then serve as the basis for the Pacific Theater 35 years later.