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The June 24, 1861 Graduating class of West point was 34 in number. That is barely sufficient for just one of the 8 company, 400 man, infantry regiments of the Civil War Union Army. Even assuming that a full 40 years of graduates were available that would be barely cover the unit needs of 40 regiments, or 40 * 400 = 16,000 men. That is but one eleventh of the officer needs for the 175,000 men at Gettysburg in July 1863.
Add on that the officers were split between two warring sides of a few hundred thousand men each, that suffered extensive casualties and had additional staff officer requirements, and one would not be far off to say that the officer corps for both sides was, for practical purposes, nearly 100% amateur. Further, even at the end of the War it is likely that few of the officers had even a full three years experience either in uniform or in combat tour.
By way of comparison, the American Civil War is not even as long as the War of the First Coalition, 1792-1797. That latter saw the recognition of numerous future French military leaders, foremost among them Napoleon, yet most if not all of the future Marshals of France were yet to be promoted past Brigadier General.
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Trained Officers averaged approximately 1 per regiment
I know I'm answering an old post with an already-accepted answer, but there was only one and it missed out on a key feature of the War: Military colleges. The US didn't just have West Point for trained army officers. In the United States before the war, especially in the South, there was a sense that a fight was coming. In the 1840s numerous state military colleges were established which provided students a military education roughly on par with West Point. I say roughly because for the most part West Point did (and still does) focus on engineering, so a graduate of a State military school like VMI might get the same infantry/artillery/cavalry training as a West Point grad, but by and large missed out on the Civil and Military engineering aspects.
In "Seed Corn of the Confederacy" by James Gindlesperger it's stated that at the start of the war trained officers (those that graduated a military college of any sort or had served as an officer in the US Army or foreign army) averaged about 1 per 1000 men. As a Civil war regiment nominally contained 1000 men (though in reality as they weren't reinforced 400 was closer to an accurate average size) you get about 1-2 trained soldiers per regiment. At the start of the war this number favored the south, as their more numerous military colleges provided a larger pool of nominally-trained officers. For example the famed "Stonewall Brigade" at 1st Manassas had a high number of VMI graduates as officers (IIRC something like 30% of the officers captain and above were VMI grads but I'm away from my books so don't take that as gospel). That'd be about 4x the "normal" amount of trained officers in the unit. As the war progressed those numbers evened out as officers were killed and both sides had a crash-course in officer training.
Of course, knowing the proper evolution for company or regimental drill (something you could expect to learn at a military college) and knowing how to run a brigade/division/corps/army are two wildly different things. There were exactly 0 divisions in the US Army when the South seceded. There weren't even brigades, army organization ended at the regiment. Prior to that the Mexican-American war in the 1840s saw the US field an army of 79,000 men. But considering the senior officers on both sides (Lee, Grant, Longstreet, etc) were mostly junior officers and none, IIRC, commanded more than a regiment during that war, the experience of anyone in command of a brigade or higher during the Civil War was nil, and it was all learned more-or-less on the job.