I am asking this question because, in my limited experience with so-called "academic" history, the United States transforms into a world power fairly quickly. Initially, I thought it happened solely because of WWI and its aftermath. However, some reading about Theodore Roosevelt's actions preceding WWI (stoking the Spanish-American War, the show of naval power) has changed my mind. The United States seems to have become a "world power" during the 19th century at some point.
I am not sure how the Civil War fits into the conversation. I know that it does, but most of my knowledge regarding the Civil War has come from rather sterilized sources (I was obsessed with Abraham Lincoln as a kid and know the little kid version of that time period). Thus, I submit this question.
Teddy Roosevelt's actions primarily concerned expansion of the US Navy - most notably the creation of the "Great White Fleet". That said while this was a modern fleet capable of defeating the Spanish navy, it was still completely outgunned by the Royal Navy. Moreover, the dreadnought race further left America behind in the battleship count, as the US Congress and the Secretary of the Navy (Josephus Daniels) had curtailed funding for the expensive new battleships. That said, given the destructive results of the naval arms race between Germany and Britain, it can be said that America's refusal to engage in such an arms race benefited the nation in the long run.
The army by contrast had in fact been largely demobilized after the Civil War - retaining only a handful of regiments for policing the Indian frontier; which is why it found itself reliant on volunteers and antiquated rifles during the Spanish-American War. It was largely demobilized again after the Spanish-American War and Philippine-American War, which is why so much of the American equipment in the First World War had to be supplied by the French.
America's "power" in the 19th Century was in fact economic. America was the world's largest producer of raw materials (particularly oil) and industrial goods by the end of that century. It didn't need to have large armies to sustain its economic power and prosperity; and indeed it was American's anti-militarist and anti-imperialist elements that saved it from getting embroiled from much of the First World War in the first place.
Unfortunately there's still this school of history (primarily British) that likes to pretend national power is an outgrowth of imperial glory to try and pretend that European imperialism+militarism wasn't so bad and that America was in fact weak in the 19th Century because it largely avoided imperial and militarist entanglements. In reality America was an economic superpower by the end of the 19th Century no matter the size of its military; and that economic power could turn into military might fairly quickly when national security was really threatened.