The simplest answer is that you'd buy your guns from a gunsmith or other merchant who would sell or import guns, but even that varied by colony and region. Even though the British, when "the troubles" started up, were understandably more sensitive to colonists' access to firearms, it was never seriously considered to attempt to prevent colonists from arming themselves. The capacity for bearing arms was considered an Englishman's right, and the history of colonial struggles and wars against hostile natives, and most male colonists of military age were expected to be able to serve in the colonial militia. Service in the militia necessitated the ownership of a firelock, a cartridge box, and a bayonet - these were unambiguously military weapons, not simply hunting weapons.

I'm going to hop around your questions a bit as I ramble here. Service in the militia, like I said above, required a man to own what was called a stand of arms - musket, cartridge box, bayonet, and a certain amount of ammunition - and, very likely, a uniform mandated by the colony or city or region in which they served. While there was some variance depending on the organization of the regional militia, a man was likely to be required to own a smoothbore musket capable of mounting a bayonet. It's possible that some companies were organized around using rifles, which, typically, could not mount a bayonet, but that was mandated by the company in which this hypothetical colonist served. If he showed up to his rifle regiment with an aging smoothbore musket, he could be fined and dismissed, or chastised in another way.

Anyway the standard longarm was a smoothbore musket that fired a .60 to .69 caliber lead ball weighing somewhere in the neighborhood of 350 grains or more. Rifle balls were a little smaller, and the typical "Pennsylvania" or "Kentucky" longrifle fired a .50 caliber ball. Apart from the inability to mount a bayonet, the rifles common to colonial rifle companies was more or less equivalent even to British military rifles. There was a totally negligible difference between the colonists smoothbore muskets and those of the British. In fact, it was extremely likely that in 1775 most American colonists serving in a militia (or at least armed as such) would have been using a Long Land Pattern Service Musket, or in common parlance nowadays a "Brown Bess" type 1. There was a chance that they'd have a French model "Charleville" type musket, and the differences between the two were, once again, negligible. The Charleville fired a smaller ball and was a little lighter than the type 1 Bess, but by about 1778 the British regiments mostly had moved to the type 2 Bess, the Short Land Pattern Service Musket. Which was, as you can guess, shorter and lighter than its predecessor, but fired the same ball of the same caliber with the same powder charge.

If, however, one wanted a rifle or fowling piece or some other type of non-military style firelock, most gunsmiths carried a huge variety of weaponry. Pistols, both rifled and smoothbore, could be bought singly or as part of a set of two(usually intended for duels). Simpler fowling pieces (essentially shotguns) were very common. Sport and target shooting rifles, double-barreled flintlock pieces, smaller, personal pistols and all sorts of other things were, if not common, at least available for purchase, special order, or custom design. Most of these would have been imported, and since the colonies were in a very particular sense meant to be a captive market for finished European goods, this relationship was encouraged and reinforced, political troubles aside.

But anyway, your question is more about how someone without a firearm would acquire one, for... reasons >.>

This is where the Committees of Safety come in. The local government infrastructure generally assumed many of the costs in arming and equipping militias, because a militia was meant to protect that infrastructure, but the idea was that if, say, the Boston government were to purchase 500 muskets for the militia's use, each militiaman would purchase a musket through the Boston city infrastructure.

Obviously, when you've got political tensions and it looks like the militias might be misbehaving, this relationship might be untrustworthy, to say the least. So, what's a good republican (small-r) to do?

Form a counter-government! Committees of Safety were essentially colonial governments-in-opposition and were the more war-centered aspect of the colonial counter-governments. Others included Committees of Correspondence, Inspection, and Observation. They can't all be neatly separated from one another, but part of their business was making sure that men friendly to the cause had access to firearms, ammunition, and other accouterments of war. The only thing that would change would be that our hypothetical soldier would purchase his arms from the Committee of Safety rather than the governor-approved militia commander.

When the war started, the continental congress assumed a lot of the duties of importing and distributing arms to the militias, as well as the continental army itself, and imports from France became much more common. The Springfield Armory, established in 1777, began manufacturing American copies of the Charleville, and by the end of the war those were the de facto standard for the US military establishment.

This is a very generalized answer and as always I'm happy to answer any follow-ups.

tl;dr One would either purchase a firelock from an import merchant, gunsmith, or other local distributor prior to the outbreak of hostilities. Before and afterward, Committees of Safety tried to import and stockpile arms for the struggle, and by the time the war was in full swing the continental congress ensured access for the army and militias to firearms.

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Some of the classic works of the War for Independence that tough on this are

Charles Royster, A Revolutionary People in Arms

John Shy, Toward Lexington

Robert Gross, The Minutemen and their World

and David Hackett Fischer, Paul Revere's Ride

Slightly more current is Thomas Allen, Tories

and one that covers the post-war material culture and its change, Kariann Akemi Yokota, Unbecoming British

More on the 2nd Amendment and American militia culture are Lawrence Cress, Citizens in Arms, and Saul Cornell, A Well-Regulated Militia

This is earlier in the 1700s than you're asking about, but while you're waiting for a answer you might be interested in this thread:

How common were guns/ammunition in America during the early 18th century? by /u/TRB1783, /u/reginaldaugustus, and a deleted user