This is a topic I post about frequently, so in addition to my response below, you may wish to check out some of these older threads:

What did "well-regulated militia" mean?

How organized was the US militia?

did the 2nd amendment cover bombs and cannons?

How did the militia change to the national guard?

How can I get hold of a firearm for the American Revolution?

How were firearms manufactured in the 18th century?

Additionally, I've written responses to a couple very similar questions in the last week or so:

How might I buy a cannon in the 1790s?

What did "well-regulated" mean, redux. There was a long conversation in follow-up threads, as well.

All that out of the way, let's get some quick answers to your questions.

Are there restrictions with respect to class, race, and gender in purchasing firearms?

It would vary from place to place, and probably shop to shop. There was an embodied - meaning armed and uniformed and mustered for service - militia of free black men in Detroit in the early months of the War of 1812, and black men certainly participated in the "troubles" before the War for Independence, and served in militia companies and in the continental line. Many of them would have had to have purchased their weapons privately before they joined.

Property ownership was an important idea in political philosophy, and it spoke to a man's "interest" in a local community, and, generally, acquiring property was not the comprehensive barrier it might seem. Plenty of black men and women, white women, and indigenous men and women were able to acquire firearms. Both US and British fur traders sold firearms, even if there were occasional attempts to stop the practice that never quite stuck. This was a longstanding practice that had gone on from the very first trade relationships between European colonizers and indigenous communities. Guns always sold and were a high priority trade item. Attempts to regulate or stop the trade were almost always ignored or circumvented; the short-lived renegade colony of "Merrymount" was initially successful because the group of traders who lived there freely sold guns and liquor to the local indigenous communities when that had been banned from proper, legal trade posts.

An extremely common element in many treaties with indigenous entities was the supply of replacement parts and access to gunsmiths for repair of weapons they already had. So it would be trivial for an American Indian to get hold of a firearm, they just might not try to order one from, say, a Boston gunsmith, they would just buy or trade for one at a trading post.

Is there an age limit?

Nothing I've ever come across suggests that there was any interest in imposing a legal restriction on the ability to purchase a firearm by age, no.

Are there special gun stores, or would I be going to the local general store?

Yes. There were gunsmiths who could make just about anything you might want, as long as it was within their skill and knowledge to do so, but almost any trader with a relationship to European imports could order anything available in Britain or the continent, as well. There was also a thriving cottage industry that produced popular weapons like the Kentucky/Pennsylvania rifle you could likely find in nearly any village or town in America.

What kind of selection might I find? Just smooth-bore muskets, or are pistols and rifles also available?

Apart from military weapons (generally smoothbore muskets capable of mounting a bayonet), you could purchase or order "fowling pieces" (hunting shotguns, basically), rifles (like the aforementioned Kentucky/Pennsylvania rifle), and numerous styles of pistol, from tiny pocket pistols, to cased dueling pistols, to large-caliber military pistols meant for cavalry use. You could also buy muskets or rifles with multiple barrels, various kinds of custom or experimental breechloaders, just about anything a gunsmith was capable of producing.

Is there anything for sale that I can shoot multiple times before reloading?

Mostly this would be the "multiple barrel" approach. There were some few repeaters around in the late 18th century, but they were generally quite complicated and expensive, and saw very little widespread use. A double-barreled shotgun, rifle, or pistol was certainly feasible, though.

How much might a firearm cost compared to the average wage in my community?

Highly dependent on wage, and the type of firearm you might want. I don't have access to cost breakdowns or average rates of pay at hand, unfortunately, but for what it's worth I've never come across a complaint that the cost of a firearm for militia service was too high, though I have frequently come across complaints that the uniforms were too expensive, which is interesting.

Can I buy this weapon on credit?

You could buy just about anything on credit. Credit was a ubiquitous social and economic institution by the late 18th century, but much of it would reflect your social standing rather than your income. So it sort of circles around the same issues of citizenship and duty to serve the local community that militia requirements reflect.

Almost forgot. How much does ammo cost?

Again, I don't have access to any resources that break this down. Ammunition and access to it was a considerable problem for the Continental Army throughout the War for Independence, and domestic production was always supplemented by import in the US for decades after independence. Lead for shot was always cheap, though, and it's likely that the cost of powder would wax and wane like any other good.

I don't know if this is allowed, but I just wanted to thank the mods for removing the clearly incorrect politically motivated answer that was up here earlier. I want you folk to know there are readers who appreciate your work.