Did Washington, Adams, and Jefferson face problems internationally?

The short answer to this question is yes, and broadly speaking, the main challenges they faced stemmed from the nearly forty-year European war precipitated by the French Revolution. Let us look at the challenges faced by these presidents individually.

The French Revolution broke out very early in George Washington's presidency, and after the French revolutionaries found themselves at war with the monarchies of Europe, especially Great Britain, Washington issued a proclamation asserting that the United States would remain neutral in the conflict. Washington faced pressure from some factions in the United States to side with France, and his position was tested when a French officer and diplomat named Edmund Genet attempted to recruit Americans to serve in the French revolutionary army. Also, British ships seized American merchant vessels bound for France, impressing, or forcing into service, some of their crews. Under Washington, John Jay, a special emissary, concluded a treaty with Great Britain that essentially guaranteed that the United States would not go to war with that nation. This very unpopular treaty provoked French ire, which the next President, John Adams, had to contend with.

The French navy responded to Jay Treaty by doing more or less the same thing the British had been doing--seizing American ships on the high seas. Adams, the new president, sent three representatives to France in 1797 to try to negotiate an end to the depredations, and when the French demanded a bribe in return for peace talks, Adams responded by committing more funds to building up the American navy. This "XYZ Affair" led to a "Quasi-War" on the high seas, as French ships (and some British ships too) attacked American ships. 

Thomas Jefferson attempted to deal with a blockade of European ports by Great Britain, and a corresponding edict from France that all ships trading with the British would be captured, by levying an embargo on all trade. This was a disastrous measure, though it did avert war with Great Britain for a time. Jefferson was more successful in other foreign policy ventures, using the navy to stop the depredations of the Barbary Coast pirates on American shipping, and, more important, presiding over the purchase of the Louisiana Territory in 1803. 

James Madison's presidency was punctuated by outright war with Great Britain, the so-called War of 1812. It was fought in no small part due to continued impressment of American sailors by the British navy, and also as a result of the perception that British agents were arming Native Americans to attack the American frontier. The war ended as essentially a draw, and was a crucial step toward ensuring America's survival as a nation. 

It should be noted that these brief synopses do not include Indian relations. Dealings with Native peoples, including the Ohio Valley Indians and the powerful Creeks, were also key aspects of American foreign policy.