How and why did the U.S refashion its relations with other nations in the early nineteenth century?
US foreign policy during the early 1800s was largely a reaction to our experiences and fears from the 1790s. As a young republic, we were easily ignored, threatened, exploited and our sovereignty unrecognized and violated. Britain seized our ships and sailors, while France seized our shipping. Pirates taxed us in the Mediterranean, while our border with British Canada remained controversial.
So in the first decades of the 1800s we took measures to protect our independence (War against the Barbary Pirates, War of 1812) as well as measures to keep us out of the entangling alliances Washington warned us about. The Napoleonic Wars of the early 1800s involved nearly all of our best and biggest trading partners, and there was natural pressure for us to choose sides. We resisted with the Embargo Act and the Non-Intercourse Act, which made us unpopular both at home and abroad. But it did manage to keep us out of the war.
In the very early 19th century (and for that matter in the late 18th century) the United States was trying to refashion its relations with other countries in a couple of ways.
First, I would say that it was trying to assert its independence more vigorously. This could be seen in its actions towards the Barbary Pirates and towards the British and French when they disrupted US shipping. The US did this so as not to have its rights continuously trampled by other countries.
At the same time, the US was trying to avoid alliances. This was because it did not want to be dragged into the Napoleonic wars. You can see this in the way that the US tried to remain neutral (for example through Jefferson's Embargo Act) during the wars between the British and the French.
American leaders fashioned important new relationships with the outside world. Fallowing the war of 1812, the United States ended its neocolonial dependence on England and Europe, while the Monroe Doctrine defined a portentous new relationship with the emerging nations of Latin America.