How did the U.S. expand itself imperially in the late 19th and early 20th centuries and why interested in obtaining control over foreign...

The motivations of American imperialism were complex, but essentially grew out of the lack of space for "Manifest Destiny"-type expansion once the West Coast was reached, ethnocentrism and the desire to share the "superior" institutions of American-style democracy.  Economic pressures were also involved, of course, and the same imperialist desires which motivated Great Britain, Rome and all previous empires.

When America's Manifest Destiny ran right to the edge of the Pacific Ocean, many economic and political interests in the US came to believe that the country's future lay in expansion on the Pacific Rim.  Hawaii and Alaska (including the Aleutian Islands) became American territories, American trading interests in China expanded and then "opened" Japan.  Trade had always flourished in the Caribbean basin and with Latin America, and in the 1890s the plight of the population of Cuba and their insurrection against Spain captured the attention of the American public.  The draconian measures of Cuban Governor General Weyler caused a public outcry, exacerbated by the newspapers of W. R. Hearst.  The sinking of the USS Maine, accidental or otherwise, became the final cause of war between the US and Spain in 1898.  Following the successful conclusion of that war America found herself in possession of vast overseas territories including Cuba in the Caribbean and the Philippines on the edge of the Pacific and the China seas.

This situation was seen by many economic interests and politicians as the natural progression of the US into a great power, but was considered an embarrassment by many who believed the foundations of our republic were threatened by such imperialism, and had no desire to become a colonising nation.  The hypocrisy of an imperial republic became particularly acute in the campaign America undertook to "pacify" the Moro population of the Philippines, who had always resisted Spanish rule and now found themselves under the "benevolent" control of the US.  In many respects this led to the isolationism espoused by many Americans throughout the first half of the 20th century.  So the main methods of American imperialism were economic expansion and war, the twin engines which have powered imperialism throughout history.