For the Union, and specifically Indiana, I found this very useful analysis:

 The training the Civil War soldier received was quite different from that of recruits in modern armies. After entering the Army, by far the most common form of training enlisted men received was marching drills.  Soldiers’ letters and memoirs are filled with references to drilling.  Men were taught how to march in columns and deploy into battle lines.  Theodore Upson recalled how his regiment was also instructed in how to form a hollow square: a configuration in which infantry could defend against cavalry by having the men fix bayonets and form a square with all men facing outward.  There appears to have been very little in the way of training or conditioning beyond drilling.  One rarely finds descriptions of target practice.  Wayne Alford found himself deployed to Kentucky without even having been issued a weapon, let alone firing one.  Beyond marching, there is no evidence that men did any significant physical training, underwent training in hand-to-hand combat, or were put through live fire exercises to acclimate them to the noise and smoke of battle. 

Essentially, little practical training was given, and what was given focused heavily on drilling and formations. They probably learned their drills very well but whether or not these were applicable in a battle is another question. It was very different from modern "boot camp"/training.

Source: Hoosier Soldiers in the Civil War by Thomas E. Rodgers https://www.in.gov/history/3991.htm