The Regular troops wore the regulation uniform of the US Army. That included the "uniform" and the "fatigue dress."
The full uniform, of infantry was this:
And there was fatigue dress, including a forage cap and blue flannel "blouse" in lieu of the coat:
it was customary (but not universal) in the period, for the troops to wear fatigue dress on campaign.
The Regular Army soldiers recieved a full clothing allowance from the Army each year. If the soldier was thrifty and handy with a needle, etc. he might not draw his full allowance, in which case he was paid for clothing not drawn. If his clothing was lost or destroyed through negligence, and he had to overdraw his allowance, it came out of his pay.
The principal distinction between the Regulars and Volunteers was, the Regular Troops received ALL of their clothing from the US Army.
The Volunteer soldiers of the US Army were due the same pay, and allowance for clothing as the regulars, but there was no law which specified they had to wear US Army clothing. Many States supplied their own volunteers for a time, with the federal government reimbursing them.
However, by mid-war, the Army Quartermaster system was capable of supplying the wants of most of the army for the most part. Volunteer units, generally, did not bother to draw the full uniform and fatigue dress. Only what they needed at any given time.
John Billings, in Hardtack and Coffee:
Only on March 3, 1865 was it directed by law that Volunteer Troops be actually ISSUED the FULL Regular Army clothing in Kind (not in clothing money).