The French Revolution was in many ways an outgrowth of the American Revolution, based on many of the same democratic ideals. France had also been a valuable ally in the American Revolution, without which victory against Britain might have been impossible. For these reasons, Americans were broadly supportive of the French Revolution initially.
However, as the revolution dragged on, the instability in France got worse, and some of the revolutionaries in France were too radical even for the American revolutionaries.
Attitudes toward France became linked to existing political divisions in the US.
Thomas Jefferson's Democratic-Republican Party was in favor of the French Revolution, based on its democratic ideals.
Alexander Hamilton's Federalist Party was largely opposed to the French Revolution, due to the disorder it had created. Hamilton was also interested in restoring ties to Britain as a valuable trading partner.
In the 1790s, the French became increasingly radical and several monarchic countries in Europe (including Britain) rose up against them.
American sentiment turned against France when in 1794, during the infamous Terror, the French arrested and detained Thomas Paine, who was an American citizen and an important activist. The Federalists took control of the US government and severed ties to France, resulting in a series of diplomatic incidents culminating in a cold war with France called the Quasi-War.
A few years after that, Napoleon seized power over France, and hardly anyone in the US was willing to support France anymore.
The French Revolution provided both idealistic and practical problems for Americans. The ideals of the French Revolution had been spawned in part by the American Revolution. The slogan of "liberty, equality and brotherhood" closely resembles America's " life, liberty and the pursuit of happiness" mentioned in the Declaration of Independence. So, Americans were very sympathetic towards the French people, who, like the Americans, were fighting to overcome a political system, lead by a king, that paid no attention to their views. However, by supporting the French Revolution too closely, America risked going to war with England and their allies who were terrified that the revolution would spread to their countries. As a new country trying to establish itself, most American political leaders knew another war would bankrupt the country. Consequently, many Americans supported the French Revolution with their hearts, but supported the British with their heads.
The French Revolution served as a litmus test of sorts in early American politics. With the American Revolution only a few years removed from its conclusion, many Americans were sympathetic to the French desire for liberty and freedom from monarchal rule. Thomas Jefferson was a leading French sympathizer in the United States and as events grew increasingly violent in France, Jefferson faced fierce criticism for his support of what his opponents saw as mob violence and radicalism. His opponents, such as John Adams and many Federalists, would eventually favor support of the British in their effort to suppress the revolutionary regime of Napoleon. They saw the French Revolution very differently than their own efforts against British rules a few years earlier. The French Revolution seemed to be a manifestation of all their worst fears about revolution: chaos, disorder, and strong-man tyranny.