The U.S.'s first large-scale overseas war?

Depends what you call "large scale".

The first overseas action of the US armed forces (if you take overseas to mean outside the North American continent) was the raid on Whitehaven in the US war of independence (1778): https://en.wikipedia.org/wiki/John_Paul_Jones#Ranger_attacks_the_British
Having visited Whitehaven, this is not forgotten.

The first war the US engaged in after independence was also "overseas", against the "Barbary Pirates" of what is now Libya (18'1-1805): https://en.wikipedia.org/wiki/First_Barbary_War
Ships from the North American colonies in the Mediterranean had been protected by the Royal Navy prior to independence. This was no longer the case, and resulted in war. Hence the line "Shores of Tripoli" in the US Navy hymn.

The war with Spain in 1898 was relatively minor compared to US involvement in Europe in World War I. It did have significant impact, sure - but in terms of man and material, it was peanuts compared to World War I. We didn't even have a draft for the Spanish-American War, just a volunteer military.

The Spanish American war may have been minor in comparison to the "Great War" in Europe. However, it was America's early taste at global expansion with the taking of the Philippines and the even Guam in the Pacific in addition to Caribbean (Puerto Rico) conquests. While the Filipinos achieved independence following the second Great War, the US still has Puerto Rico and Guam. Some Americans don't understand that Puerto Rico and Guam folks are American citizens.

Did you spend more than one day on the Spanish-American War in your high school history class?

Some Americans don't understand that Puerto Rico and Guam folks are
American citizens.

... whose head of state is the U.S. president, in whose elections they are not entitled to vote.