What happened to the confederate soldiers when they surrendered?

Mostly they were considered captured and "paroled" and sent on their way home. A "parole" was a promise by the captured man to return to his home and not fight any more until "properly exchanged", by a cartel of officers from the two sides who met to "exchange prisoners", by using names from lists of those captured. Nobody was entirely sure the war was over at the several Confederate surrenders, even those which took place after Lee's. Often the Rebels were given a little food by the Yankees to start them on the way home. Mostly they had to walk. In Lee's army, Lee negotiated with Grant that any Confederate soldier who claimed to own the horse he was using could keep it. Most Union horses were government property, but Confederate cavalrymen had to provide their own horses.