Every federal trooper was armed with a revolver and a saber. It was not however until after campaign season of 1863 had been completed that Union trooper was issued a long arm. Ironically on the southern side, sabers were always in short supply.
The self contained cartridge did not make its appearance in any significant numbers until the last year of the war, with the issuing of the Spencer carbine firing rim fire .56-50 cartridge. The breechloaders prior to that and for the most part were fed ammunition in the form of a bullet tied to a combustible paper tube containing the powder charge. The Smith carbine used a form of India rubber tube to hold the powde, the Maynard, Burnside and Gallagher carbines all used a spun copper or brass cartridge case rather than paper but theses were supplied in fewer numbers than the sharps cabine. Never the less, all were externally primed with the ubiquitous winged musket cap.
Yes it is true that many a confederate trooper found himself armed with multiple revolvers, one must keep in mind, they were short ranged weapons. While capable of great accuracy out to 25 to 30 yards they were rarely employed this way. Typically they were used up close just as they are today.