The US Army traces its history back to the formation of the Continental Army on 14 June, 1775.
There are Army wide traditions, and unit level traditions. Some of these are very serious. We trace the civilian control of the US Armed Forces to George Washington's voluntary release of power after the American Revolution. The strong tradition of following the orders of our civilian leadership is also an important part of the culture of the whole nation. Other traditions are very local, such as the rights of the Third Infantry to march with fixed bayonets.
Many units in the US Army have strong traditions such as unit mottoes. When an enlisted man salutes an officer in many of these units, he will call out the unit motto, such as "Can Do!" or "Twenty Rounds Full."
There are traditions that have been allowed to die. These include a deliberate separation of the soldier from society. Some have mutated. The old cavalry tradition of "Horse, Saddle and Rider" was about the order in which a trooper was to take care of matters. While we don't have many horses left in the service, we still expect to take care of our vehicles, then our personally assigned gear and then ourselves, in that order.
Then there are traditions that still exist, and are scarcely noticed. Golf is one of those. Prior to WWII, the officer corps had more than its fair share of wealthy members. They liked golf, but in order to justify golf courses on Army land, more had to play golf than just the wealthy. Golf was encouraged as a healthy pastime for the whole Army, and as this gradually seeped into the civilian world, it ceased to be a mark of how unusual the army was in fondness for golf.
There are traditions which have become famous from the movies, such as the Rangers with "Leave no man behind" in the Ranger Creed. Exposure to this has led to many soldiers thinking this is just the normal way of doing business.
However, what is certainly the most famous tradition of the Army is calling cadence. This spread to other services and nation and even outside of the armed forces. A cadence is a beat that helps soldiers perform a task in unison. Originally this might mean marching or loading a musket. The modern use of the term refers to a chant credited to a private Ducksworth in 1944, in which each time the soldier's left foot hit the ground, he would repeat words called out by the soldier who was leading the cadence. In the original case, it was a chant that started with "Sound off! One, two, Sound off, Three four" with variations in timing and content to keep interest up. It was a means of inspiring tired troops into a bit more enthusiasm. Now there are hundreds of elaborate cadences, some of which are tied to Army or unit traditions, others of which are common throughout the Armed Forces. Seldom will anyone watch a military movie without some sign of cadence being called.