which divisions, unit, corps, etc. was in charge of running [training]

ALL OF THEM, more or less. U.S. Army training was very decentralized.

So in 1940, when the U.S. Army began preparing in a serious way for war, it started from a very small peacetime establishment. Consequently, the entire U.S. Army was one big training program from 1940-42 and even while small numbers of U.S. Army divisions were fighting between late '42 and D-Day in 1944, most divisions were not in combat and still completing their training program.

Since the Army was focused on training, ultimately it's General Headquarters (GHQ) which was later renamed to Army Ground Forces, was in command of the training program. As GHQ/AGF commander, Leslie J. McNair was the officer most responsible for the training and mobilization plans of the U.S. Army. He designed a training and mobilization scheme in which individual combat units were responsible for training their own personnel.

The way this process worked is that enlistees or draftees were assigned directly to their division. Each new division started with a cadre, a group of more experienced officers and men who were responsible for training the new recruits, all the way from basic training up to battalion and division level maneuvers. However, once a division was making progress in becoming trained, it often became subject to personnel raids, especially for Cadre for new divisions and occasionally for trained men to send to divisions already in combat as replacements. Men were then promoted to fill the gaps and newer draftees were added to replace them and the training cycle would begin again to bring them up to speed. This process in it's early phases, 1940-3, rather resembled cell division. Later in the war, when all the divisions were activated there were more specialized programs for training replacements, and the last divisions to see combat frequently trained very large numbers of men as replacements for the older divisions already in combat.

As more and more divisions entered combat, they began to need replacements for casualties suffered. These men were sometimes trained under the Replacement and School Command setup in 1942. However this organization was initially not up to the task of training the replacements needed and they were often stripped out from newer combat units instead. Only later in the war did a well organized replacement training system with the infamous "Repple Depots", Replacement Depots begin to emerge.