Nineteenth century American imperialism got off to a bang in 1803 with the Louisiana Purchase. Napoleon initially wanted to revive the French empire in the Western Hemisphere, but the cost of crushing a revolt in Haiti and more pressing needs in Europe persuaded him to abandon the scheme. The Native Americans who actually occupied the land in question were not, of course, consulted in this matter.
One of the American objectives during the War of 1812 was the conquest of Canada. This part of the war was a miserable failure. The war, as a whole, was a draw; the Native Americans, however, were the big losers, as the British quietly gave up on their plan to create a buffer state for Canada that would be controlled by the natives. American campaigns against Native Americans in nominally Spanish areas that are now included in Alabama and Florida also set the stage for the acquisition of these areas after the war.
The Monroe Doctrine was inspired by British efforts to prevent the restored Spanish monarchy from reasserting control over its former colonies. America had no military power to prevent European intervention in South American countries, so it essentially had to rely on the British fleet for that purpose. The Doctrine was not, therefore, an attempt to establish American hegemony over the Western Hemisphere, as no resources existed to make that a reality. It was, in reality, a blow against autocracy and imperialism, not an attempt to assert an American variant.
The highlight, if you could call it that, of American imperialism during the first part of the nineteenth century was the Mexican War, which was provoked by President Polk in an effort to force Mexico to sell, at a minimum, its northern possessions to the United States. The war was strongly opposed by the Whig Party, but it was a huge military success. Politically, it was a disaster; the acquisition of new territory made the Civil War much more likely.