Another question I have is Would soldiers be issued a fatigue uniform and a dress uniform or how did that work?
The Regular Army of the United States provided each soldier a uniform, and fatigue dress.
The uniform looked like this...
the fatigue dress was this, a forage cap in lieu of the uniform hat, and flannel blouse in lieu of the uniform coat etc.
Most US troops took the field in their fatigue dress...
The regulars were given an allowance of clothing, from the table shown in a previous post above. IF they had to draw more than what the table allowed, they were charged for it from their pay.
The Volunteer troops that composed the bulk of the Union army were not issued clothing like the regulars. They were allowed clothing money, at the same cost as the clothing allowance to a regular soldier. Most Volunteer units, after 1861, just purchased US Army clothing, particularly the fatigue dress. There were some units that insisted their men procure dress uniforms, but that was an exception rather than the rule. Any part of the clothing money not spent the soldier would evidently get in cash at the end of the year, etc.
Unlike the regular troops, who would be punished for discarding their clothing and equipment, the volunteer troops did so at will, as they technically paid for it, etc. But according to John D. Billings, the men and their officers were chock up the losses to "battle" and claim gratuitous replacements from the quartermasters...
The clothing money system did not survive the war.
A law of March 3, 1865 directed all volunteer troops in federal service would thereafter receive the same amount of clothing, in kind, as enlisted men of the same arm in the regular army.