In what ways did the United States seek to influence other nations between 1872 and 1917? How was the United States changed by its relationship with other nations during this period?

Your question extends across a long timeframe, beginning with Reconstruction and ending with World War I. In this time period, the United States had advanced upon large scale industrialization and made a move towards adopting a more imperialist foreign policy. By the end of the time in question, it had gotten itself swept up in one of world history's largest and most destructive military conflicts.
For much of its history, the United States had tried to stay out of European affairs, a position which was perhaps most clearly expressed in the Monroe Doctrine. United States expansion, throughout the first half of the nineteenth century, was entirely continental, as the United States spread from the Atlantic towards the the Pacific. As the United States entered into the industrial age, however, it embarked upon a new era of foreign expansion—this time it was overseas.
It's important to note that imperialism and industrialization are very closely intertwined. Capitalist and industrial economies tend to push for colonization to acquire raw materials as well as access to foreign markets. Therefore, it should not be surprising to find that with industrialization came a change in foreign policy. The interests of Big Business pushed for foreign intervention and so did the writings of military theorists like Alfred Thayer Mahan. The United States, by the close of the century, had modernized its navy and entered war with Spain, gaining Spanish possessions in the Philippines, Puerto Rico, and Guam, as well as setting up a protectorate in Cuba. In doing so, it vastly increased its own economic, military and political reach.
With this came an evolution in foreign policy. Dating back to the Monroe Doctrine, the United States had, for all intents and purposes, defined the entire Western Hemisphere as an area of American interest, but with the turn of the century, we see a new intensity of United States interference in Latin America. Theodore Roosevelt famously issued the Roosevelt Corollary of the Monroe Doctrine, which set the United States up as an intermediary across the entire Latin American world and, later, policies of Dollar Diplomacy would only further extend American claims of hegemony throughout the Region.
At the same time the United States was expanding its influence within Latin America, it was also expanding its influence across the world. In the interest of accessing markets in China, the United States attempted to push Europe to accept the Open Door Policy. It also contributed troops to the suppression of the Boxer Rebellion. Later, in 1905, President Theodore Roosevelt negotiated peace talks between Russia and Japan in order to end the Russo-Japanese War. What we see across this time period is the United States becoming more interventionist, acting as an imperial power to assert its own interests across the world.
By the end of your timeline, Europe is in crisis, swept up in the first World War. I find it interesting that it ends specifically in the year 1917, which was when the United States entered the war. Before this point, the United States had charted a course of neutrality, with opinion divided within the country as to which side the States should take. In any case, you should be aware that the experience of World War I would serve as a key turning point in global history, and that it created with it a severe sense of disillusionment in the decades that followed.
If we were to look towards the 1920s, we'll see the United States pushing more in line with the calls for global disarmament, and as Totalitarian States rose to power, the United States would respond with a much more non-interventionist stance—all the way until the attack on Pearl Harbor.