For context, I'm currently studying the history of the Second Amendment. It seems that there was a direct link between "keeping and bearing arms" and the maintenance of a militia so as to avoid a nation-wide standing army.
So when armies were raised, did the soldiers bring their own guns with them? Or were they issued their own upon arrival? What type of guns were used during this time?
The militia was a sort of organizational ideal, rather than a consistent coherent organization, and as such its standards of armament, its uniforms, its leadership, and even its branch, were all particular to their region at specific times. However, in general, membership in a militia was restricted to citizens with a vested economic and social interest in their communities. As a part of that interest, they were meant to furnish their own weapons and uniforms according to the local organization, as well as any additional costs; if you were part of a cavalry regiment, for instance, you'd also need a horse and a sword, and if you were part of a cannon crew, you might have to pay into the maintenance costs of your cannon, and pooling money to buy powder and shot.
As for types, in order for a man to be considered "armed" in the context of militia/military service was what was called a "stand of arms" which consisted of a musket, bayonet, and cartridge box. A musket was a military firelock in the sense that something like a fowling piece (for hunting birds) wasn't, in part because it was capable of bearing a bayonet.
Muskets made by different nations were pretty comparable to one another; the two most common used by Americans in the post-War for Independence period would have been the Long or Short Land Pattern Service Musket produced by the British, or the Charleville, used by the French. Both the long and short land patterns were .75 caliber and fired a .69 caliber ball; the Charleville was .69 caliber, and usually fired a .65 caliber ball or so. They weighed within a couple pounds of each other, were comparable in length, and apart from a few nuances were more or less equal in performance.
The United States started manufacturing its own weapons in the Springfield Armory, patterning their muskets off of the French pattern. By the War of 1812, the Springfield armory was producing enough weapons to arm the (very small) US Army, but it wouldn't have been uncommon for militia regiments to use imports of various kinds.
It's important to point out that the militia was intended to act in the capacity of a standing army to repel invasion and suppress insurrection, and so the militia needed all the things that standing armies would need: cannons and cavalry in addition to infantry. Americans in particular made use of rifle regiments. All of these weapons would be purchased privately and maintained at the cost of the community.
So what about all those incredibly frequent examples of militia showing up for musters without shoes, or arms, or blankets? For instance, shortly after the outbreak of the War of 1812, after months of fluctuating militia presence along the Niagara frontier, there were fewer than 800 militia present and “many without shoes and otherwise illy prepared for offensive operations.” Even those 800 were difficult to keep, as just days later they were loudly clamoring for pay, and it was once again noted that they were without shoes.
The problem was that, without any kind of centralizing authority or central logistics depot, local regions were subject to local logistical problems and, perhaps more importantly, local politics. Some regions spent a great deal of time and money in preparing their militia and using it, and others spent much less time, and even openly protested the necessity of every citizen bearing arms. I went into more detail in this answer. In places where militia requirements were taken seriously, fines were levied for failure to muster, failure to keep or maintain your arms, and for other failures to shoulder the duty alongside your community members.
I don't want to give the impression that the militia as an institution was a failure - though it's not hard to find that opinion among historians - but rather to give the impression that the militia as an institution was about far more than military preparedness. It was as much a reflection of the social and political reality of the region as it was a measure of the organized expression of military force. Leadership was democratic, and often reflected the region's assessment of the individual captain/colonel/whoever's suitability not only as a military commander, but as a powerful, influential, wealthy, or famous person. The militia itself was omnipresent, even in rotating watch duties, ad hoc organizations to search for missing townsfolk, repel raids, apprehend criminals, fight fires, even to muster for parades and other town celebrations. Organizing in a military sense was actually a rather small element of its ideals and reality.
I've written quite a lot about this in the past, and I'm happy to answer followups.
Sources
Charles Royster, A Revolutionary People in Arms
John Shy, Toward Lexington
Robert Gross, The Minutemen and their World
More on the 2nd Amendment and American militia culture are Lawrence Cress, Citizens in Arms, and Saul Cornell, A Well-Regulated Militia
Also keep in mind that there was were three distinct types of Army soldiers: regular Army (federal), militia (state), and volunteer (federal). If it helps, think of them like the Active Duty, National Guard, and Reserve, respectively. The volunteers, unlike the current Army Reserve, would be recruited and sent to the front relatively quickly with no training and likely very little equipment. Volunteers would not necessarily have any equipment with them: the expectation was that the federal government would provide them with clothing and muskets prior to their deployment.^(1)
In terms of armament and whether or not the state required the militia to arm themselves, it depends. Several states had contracts with private gun manufacturers to provide their militias with weapons. For instance, Eli Whitney had a contract with the state of New York beginning in 1808 to provide weapons for its militia. ^(2) Additionally, Virginia supplied at least one company of Volunteers with weapons from its reserve militia inventory when the federal government could not equip them.^(3) But New York and Virginia may have been outside of the norm. Massachusetts, who had arguably the best trained militia, required citizens to provide for their own arms and would fine anyone who arrived to muster without their equipment.^(4)