One major point of strife would be the Quasi War of 1798. This was largely a naval war engaged between the United States and France.

Some of the events that led to the war was that the United States was being drawn into the conflict between Great Britain and France during the Washington Administration, through actions such as the impressment of American sailors into the British Navy. Washington wanted to pursue a course of neutrality. John Jay was dispatched to negotiate a treaty to secure neutrality with Great Britain. This ended up occurring, and the Jay Treaty was passed and signed by the Senate.

After Washington left office and Adams succeeded him, Adams was left with deteriorating relations with France. This would be further set off by the "XYZ Affair." This were three ministers that Adams dispatched to negotiate with the French being unable to meet with the French Foreign Minister, Talleyrand. The French demanded bribes, essentially, for the American minsters to meet with Talleyrand. Adams did not do so and so this led to an escalation between the United States, mostly contained to naval skirmishes.

This would largely continue until a peace was negotiated between American ministers and Napoleon shortly before the election of 1800, which Adams would go on to lose.

This is mostly where my knowledge of French-US relations ends in the time period of your question. But I would note that the Federalists, of whom Adams was one, were more generally favorable of the British and the Jeffersonian Republicans were more favorable of the French. After the 1800 election, the Jeffersonian Republicans would be in control of both Houses of Congress and of both of the Executive (POTUS and VPOTUS) seats.

Forgive any grammatical issues, but it is late for me.

Source: Ferling, John. Adams v. Jefferson: The Tumultuous Election of 1800. Oxford University Press, 2005.