How did the north and south get more soldiers?

 They drafted, or conscripted, most of their troops. For the North, it was the first draft in U.S. history and not well received (see the New York Draft Riots). The Union draft also allowed men to pay others to take their place in the draft. 

The Southern draft was enforced by the Confederate Home Guard, technically made up of men too old or too young to serve at the front. My randfather was a native of Winston County, Alabama, and was shot in the back when he refused the Home Guard's request that he report the Confederate Army. (Winston County, btw, tried to succeed from Alabama when Alabama succeeded from the Union earning it the name of the Free State of Winston.) 

Early on, both sides filled their armies with volunteers. However, as the war went on and casulties mounted, volunteers became few and far between, resulting in the draft. 

There were no mercenary units recruited by either the North or South to fill the armies. However, Union recruitment officers made a point to greet incoming immigrant ships and impress military-age immigrants into the army.