It really depends and I don't have the census data I'd need to be precise, so I can only speak in generalities. Free people of color occupy just about every position on the economic ladder from planter with lots of slaves (though this is rare) to destitute. They don't exist in significant numbers in most of the South until after the Revolution, with the notable exception of Spanish (but previously and subsequently French) Louisiana.

In the Upper South (the Chesapeake and points west, basically) free black populations become fairly large. By 1860 they dramatically outnumber enslaved blacks in Delaware and Maryland is about 50/50. Some certainly do well in freedom, but their numbers are large enough that whites see them as a serious problem and become more vigilant about policing the racial boundaries. That means they have much more difficulty getting ahead and tend to live lives much more like they and their ancestors led while enslaved. The Revolutionary-era wave of manumissions passes quickly (and tends to be oversold as evidence that they were so close to getting rid of slavery altogether) so from then on you're looking at mostly at people who are freeborn rather than ex-slaves, but manumission never completely stops despite increasing legal obstacles to it.

Among those barriers are requirements that the manumitting owner post bonds against a freedperson's good behavior and/or that they must leave the state. There's always a risk of re-enslavement as punishment for crimes, not being properly under white control, or kidnapping for sale elsewhere. Southern whites tend to find the idea of a free black person deeply horrifying, even if they occasionally free people themselves and think those freedpeople are a-ok. To be free and black is to be of low moral character: lazy, shiftless, and criminal.

In the Lower South (the cotton states), free people of color are much rarer and more concentrated in cities. Rural populations do exist, but they're less common. Substantial (in numbers and economically) communities exist in particularly Charleston and New Orleans. New Orleans is a little weird so I'll treat it as its own subject. In Charleston and environs there are some free blacks who do well indeed, though for most of them that probably means something closer to upper middle class than filthy rich. Many of these people are the sons of rich white planters who freed them and set them up in some comfort. They are not equals of whites socially or legally, but the patronage of their relations and a certain degree of personal wealth arising from it does make a difference. This is sometimes described as a distinction between "brown" (we would say biracial or multiracial, but they said mulatto) and "black".

A few rungs down from those relative fortunates, you have urban artisans who basically hack Charleston's mores to get a limited advantage over their white competition. To openly solicit business on the street (obtruding, in the parlance of the time) was degrading for a white tradesman. It implied the kind of money-grubbing that the gentlemen of Charleston considered fit only for Yankees. But a black person was already at the bottom of the totem pole and thus could obtrude without any trouble. Thus they got enough business which their white competitors thought rightly theirs that they frequently petitioned the city government to do something about the problem.

New Orleans arrives at something similar to the larger situation in the Lower South, but from a different place. Slavery boomed early on in French Louisiana, but then fell into a serious decline. The French also recognized a legal right to buy one's own freedom. Put the two together and a fair number of surplus slaves are able to buy their way out of bondage without the kind of trouble they would have had elsewhere. By the Louisiana Purchase, they and their descendants constitute a kind of middling (not exactly between and not exactly equal to whites, but above slaves) class usually called free people of color. (This is often rendered in French: gens de couleur libres, or even just gens de couleur.) They're a self-conscious group who work hard at building and keeping up their own institutions, including militia companies and some other traditional prerogatives against the American insistence on a firmer color line. (The militias really freaked out Louisiana's first American governor. At least one group was still active in 1860 and offered its services to the Confederacy, were refused, and ended up in the United States Colored Troops later on. The right to self-purchase...not so much.) The gens de couleur libres still end up relying on some degree of white patronage to get around their limited access to credit and things like that, but their families were probably a degree more self-reliant and certainly more self-made than would be the case for the less cohesive free black elite over in Charleston. They were reasonably wealthy, in general.

In both cases, the distinction between freeborn and freedpeople becomes a significant cleavage during Reconstruction. The freedpeople are generally much more enthusiastic about things like land reform, whereas the freeborn already have land and/or wealth and don't particularly want a threat to property rights.

Sources

Many Thousands Gone by Ira Berlin

Inhuman Bondage by David Brion Davis

Reconstruction by Eric Foner

Prelude to Civil War and Road to Disunion (2 vols) by William Freehling