What events drew the U.S deeper into international politics in the early years of the 19th century?

The answer to this really depends on which century you are
talking about.  The US has been drawn deeper into international politics in the early
years of all three centuries of its existence.  I am going to assume that you are
talking about the 19th century.

In the early years of the
1800s, the Napoleonic Wars in Europe got the US more deeply involved in international
politics.  As France and England fought each other, their conflict spilled over to the
US.  This happened mainly because each side started to try to confiscate US shipping
headed to the other country.  This caused the US to almost go to war with France and
then, eventually, to go to war with Great Britain in
1812.

The Napoleonic Wars, then, pulled the US into
international politics because those wars caused issues between the US, England, and
France over freedom of the seas.