What central issues drew the United States deeper into international politics in the early years of the 20th century?
Since you have tagged this with WWI and other such tags, I will assume that you are taking "the early years of the century" to extend at least until 1920.
If so, the major issue that drew the United States deeper into international politics was the issue of the freedom of the seas. This went together well with the idea of "open doors" that the US had been pursuing in China much earlier in the century and late in the 19th century as well.
One of the things that the United States wanted was for all nations to be able to trade relatively freely with one another. They wanted to be able to trade with China in the early 1900s. They wanted to be able to trade with the warring countries early in WWI. In general, they thought that there should be more openness and more freedom of trade and interaction between countries.
It was this desire that helped to pull the US into WWI and it was this desire that led Pres. Wilson to push (for example) for his 14 Points at the peace talks after the war ended.