This is stupendously false, but unfortunately, is an idea that is commonly bandied about. /u/elos_ (who I'm sure knows way more about this subject than I do) did a fantastic writeup about a lot of common misconceptions relating to WWI over at /r/badhistory.

European nations didn't know how to fight with modern weapons and were still being taught as if they were using ball and muskete

This is pretty absurd, given the fact that Europeans had left behind "ball and musket" tactics at about the same time they left behind the ball and musket. Soldiers in the Franco-Prussian War were using breech-loading rifles (the Chassepot and the Dreyse) which were more advanced than the rifles commonly used in the American Civil War, and they fought in an appropriate manner - by using loose formations, skirmishing lines, et cetera. French troops early on in WWI made excessive use of skirmishing tactics and loose order combat. They weren't lining up shoulder-to-shoulder and shooting at each other.

But when the Americans went to war, since the army had experience in the Civil War and as well as the Spanish-American War

French and Prussian soldiers during the Franco-Prussian War were using more advanced weaponry and tactics than the majority of American soldiers during the American Civil War (which makes sense, given that the Franco-Prussian War occurred 5 years after the Civil War). During the Second Boer War, the British fought against Boer insurgents in South Africa who were often equipped with the exact same sorts of rifles and artillery that would see extensive use in the First World War. The Russo-Japanese War from 1904-1905 saw many of the things that would characterize warfare in WWI - trench warfare, artillery bombardments, mass mobilization, et cetera. The idea that the American Civil War and the Spanish-American War gave US forces any special insight to modern warfare that Europeans lacked is completely unfounded.

they knew right away how to wage war in the more "modern" era.

The American Army had to be equipped with and trained how to use planes, tanks, and several sorts of artillery by the French, so I'd question the notion that America was more ready for the fighting in Europe than the Europeans were. American soldiers went into battle using tactics that the Europeans had used in the early bits of the war, and suffered high casualty rates as a result. General Pershing was a big proponent of the same sort of frontal assaults that people make fun of British and French generals for carrying out during the earlier portion of the war.

I've read a fair amount about WWI warfare, and I've never seen any evidence to support the idea that American forces performed substantially better than British or French forces on the Western Front.