How did the Union and the Confederates raise their armies?

Citizens opened recruiting offices in their hometowns, held rallies, and signed up volunteers 18 and older. Confederates wanted to join. Union had sometimes draft men because of the willingness of wanting to fight. The Confederates had to institute a draft almost a full year prior to the Union, due to the smaller population in the South.

To add to this answer, it should be noted that each state in the South and the North were responsible for raising armies. And, as noted above, both sides were forced to pass conscription laws to fill their armies' ranks.