The use of skirmishers varied widely based on the theater and year of the war.
Jefferson Davis, while he was the US Secretary of War (1853-1857,) had gotten rid of the idea of separate light infantry units in favor of the universal infantryman. During this time, the War Department adopted Hardee's Rifle and Light Infantry Tactics as the standard infantry manual. The manual does outline the use of skirmishers (p33), but in reality they were used more along the line of pickets than skirmishers used by the French army. These Skirmishers were people pulled from ordinary units, using standard weapons. The troops were so poorly trained in general at the beginning of the war so that they hadn't trained as skirmishers and were easily brushed aside.
The movements of skirmishers should be subjected to such rules as will give to the commander the means of moving them in any direction with the greatest promptitude.
It is not expected that these movements should be executed with the same precision as in closed ranks, nor is it desirable, as such exactness would materially interfere with their prompt execution.
When skirmishers are thrown out to clear the way for, and to protect the advance of, the main corps, their movements should be regulated by this corps as to keep in constantly covered.
Every body of skirmishers should have a reserve, the strength and composition of which will vary according to circumstances.
If the body thrown out be within sustaining distance of the main corps, a very small reserve will be sufficient for each company, whose duty it shall be to fill vacant places, furnish the line with cartridges, relieve the fatigued, and serve as a rallying point for the skirmishers.
If the main corps be at a considerable distance, besides the company reserves, another reserve will be required, composed of entire companies, which will be employed to sustain and reinforce such parts of the line as may be warmly attacked; this reserve should be strong enough to relieve at least half the companies deployed as skirmishers.
The reserves should be placed behind the centre of the line of skirmishers, the company reserves and one hundred and fifty, and the principal reserve at four hundred paces. This rule, however, is not invariable. The reserves, while holding themselves within sustaining distance of the line, should be, as much as possible, in position to afford each other mutual protection, and must carefully profit by any accidents of the ground to conceal themselves from the view of the enemy, and to shelter themselves from his fire.
The movements of skirmishers will be executed in quick, or double-quick time. The run will be resorted to only in cases of urgent necessity.
Skirmishers will be permitted to carry their pieces in the manner most convenient to them.
The movements of skirmishers will be habitually indicated by the sounds of the bugle.
The officers, and, if necessary, the non-commissioned officers, will repeat, and cause the commands to be executed as soon as they are given; but to avoid mistakes when the signals are employed, they will wait until the last bugle note is sounded before commencing the movement.
When skirmishers are ordered to move rapidly the officers and non-commissioned officers will see that the men economize their strength, keep cool, and profit by all the advantages which the ground may offer for cover. It is only by this continual watchfulness on the part of all grades, that a line of skirmishers can attain success.
The Union did have a separate light infantry unit from the beginning of the war. Hiram Berdan, a wealthy inventor and entrepreneur, reputed to be the best marksman in the country formed and equipped a Sharpshooter regiment (later two) for the Union. These men were specially trained, wore distinctive dark green uniforms and carried Colt revolving rifles before being given 1859 Sharp's rifles later on. 17 year old William B. Greene, then a student, qualified for the unit by putting ten consecutive shots in a ten inch bull's-eye at 200 yards.
This unit was so effective at harassing the Confederates during the Seven Day's campaign, that Confederate Brigadier General Robert E. Rodes was allowed to form his own light infantry battalion. They were first used at Chancellorsville in screening Jackson's advance as well as leading in advance of the attack on the Union 11th Corps until proper fighting began.
The use of skirmishers continued to advance throughout the war, especially by the confederates and in the Western Theater.
As far as use of weapons, the specialized weapons used by the Confederates included the Enfield P53, and the Whitworth Rifle, while the Federals primarily used the 1859 Sharp's Rifle, the 1860 Spencer Rifle, and the Henry Repeating Rifle.
Brevet Lieutenant Colonel W. J. Hardee, U.S. Army, Hardee's Rifle and Light Infantry Tactics
William H. Hastings (Transcribed by), Letters from a Sharpshooter, The Civil War Letters of William B. Greene, CO G. Berdan's Sharpshooters.