A few things to look at here.

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At the beginning of the Civil War, the General-in-chief was Winfield Scott. Scott was a good general and had an excellent record, but he was very old having first entered the United States Army in 1807. He had been a general since March 9, 1814. The issue with Scott was his age. His mind was relatively clear when he was on a good day, however there were reports in early 1861 that he was unable to remain awake during some meeting and his mobility was very limited. There was no question about whether he could take a field command and his endurance was limited for the long hours a desk job required.

Based on the Mexican-American War, there were a number of very promising officers who could be considered for commands. There were a number of issues to deal with in advance of that though. First, the US army had no pension or mandatory retirement age and so it was populated by a large amount of elderly officers. In a normal situation, colonels would be promoted to general, but the colonels were very old and several had been in the army longer than Scott. Robert E Lee, at the age of 54 was one of the youngest colonels in the United States army. Secondly, the decision was made to go with a Volunteer Army raised for the rebellion rather than building up the regular army. (The regular army did grow, but by far the majority of the army was volunteer with its own organizational structure). Thirdly, very few of the former soldiers had ever managed troops in combat and most of the ones who did managed an artillery battery or infantry company in the Mexican-American War.

The Federal side in the war had a rather large problem in that it needed to raise a very large army, then send that army deep into enemy held territories. That was a logistical nightmare of the first order. In fact, it was General Grant's assistance in recruiting and training Illinois soldiers that so impressed the Governor of Illinois that he put Grant's name on top of the Illinois' quota of generals in the volunteer army.

The CSA didn't have an existing organization to work around. It created a regular army, but it was barely sketched out and there were only a few soldiers assigned to it. Those soldiers were then also brought over to the Provisional Army of the Confederacy, which was the CSA's parallel to the United States Volunteer Army. President Davis had a prejudice in favor of West Point graduates, so he promoted them into key positions at the start. The West Point graduates all had some military experience, so this helped in terms of instilling the entire army with some degree of military knowledge throughout.

The Federal army had a number of possible options for generals in various areas. Some were based on seniority (Patterson, Sumner, Buell). Some were former military who reenlisted in the army for the war(McClellan, Burnside, Grant, Charles Stone, Rosecrans, Sherman). Some were straight up political generals who came through the state militia systems (Butler). Some were all political (Sickles, JD Cox, Logan). Since this was a monumental task to decide to who promote to general, they went with a hybrid system where the generals would be selected by either the federal government, or the state governments based on a quota tied to the amount of troops raised in different states.

AT this point, it was all theoretical. None of the options had any meaningful experience commanding troops in the field. The leading options (McClellan, McDowell, Robert E Lee, etc) were staff officers in the Mexican-American War. They knew the theories of war and military requirements, but they had no practical experience.

So, for both sides in the war, it was going to be on-the-job training.