I have been doing some amateur research on the Mexican American war of 1846 and the Spanish American war of 1898 and throughout my research I have come across the mention of the federal army and volunteer units/state units. Could someone explain the differences in equipping these units and their commands and recruiting from the time period of 1846 to 1898 as well as explain how the state/volunteer units were called up for action as opposed to the federal units?
For most of the 19th century the US Army remained extremely small in relation to its European counterparts. Factors such as American distrust of standing armies and geographic isolation from other powerful militaries meant that there was really no need for a strong army. When the US was not at war the Army was primarily preoccupied with coastal and border defense as well as frontier policing. When it was at war, the Army relied mainly on volunteers to swell its ranks, as the strength of the regular Army was simply much too small to be effective in a war against either Mexico or Spain. Many of these volunteers would be drawn from local militias or the National Guard but a large number would have had no prior military experience. These volunteers generally would have served short terms of enlistment and most would have been mustered out of service once hostilities came to a close.
In 1846 prior to the war with Mexico the Army numbered about 26,000 enlisted men and 2,000 officers. As soon as the war was declared President Polk was authorized to call up 50,000 volunteers to make up the bulk of the forces used against Mexico. That's about all I can say of the Mexican-American War as I don't really know much about the topic.
In 1898 the US Army was completely unprepared for war with Spain. Once again the Army was smaller than any European power, including Spain itself, numbering about 28,000 men once again. The US had not fought a major conflict in over 30 years since the end of the Civil War. As such basically no officer within the Army had experience commanding a force larger than a regiment. After Congress approved the declaration of war on April 20th, President McKinley called for 125,000 volunteers on April 22nd, and then another 75,000 on May 25th. It was a bit of a nightmare trying to coordinate and equip all of these volunteers as the Army had no mobilization plan and an uncoordinated staff. There also wasn't enough equipment to go around. While most regulars were armed with new Krag-Jorgensen rifles and smokeless powder cartridges, many volunteers were forced to use Springfield rifles and black powder cartridges, putting them at a disadvantage.
The actual act of getting the soldiers to Cuba proved to be a disaster as well. Tampa was the main port used to ferry the soldiers to the island and it was completely lacking in the infrastructure needed to support a large army. While it is more of a popular history book, Clay Risen's The Crowded Hour does a good job detailing just how terrible the logistic situation in Tampa was. The port was too small and there weren't enough railroads for the transport of supplies, equipment, or men (at one point there was a traffic jam on the rails all the way up to Georgia). The whole process of raising and equipping the Army during the Spanish American War was a bit of a trainwreck, and the relatively poor performance of the Army compared with the Navy proved the need for reform. In the years following the war the Army would undergo some major changes under Secretary of War Elihu Root, such as the founding of the Army War College in 1903.
Sources: Risen, C. (2020). The Crowded Hour: Theodore Roosevelt, the Rough Riders, and the Dawn of the American Century. Scribner.
Stewart, R. W. (Ed.). (2005). American military history (Vol. 1). Center of Military History, United States Army.
Weigley, R. F. (1984). History of the United States army. Indiana Univ. Press.