Furniture is a fairly expensive, yet necessary, aspect of modern life. Heck, IKEA is the "cheap" option and their prices are still fairly high for crap quality pieces.
Yet when we come in contact with surviving furniture pieces from the early 20th and late 19th century, they all seem to be extremely high quality and often decorative. Obviously, there's survivorship bias going on. But it raises the question: for a working-class or middle-class family in the United States in the late 19th century, how much would, say, a new bed or dining table cost? What if they wanted to buy a bureau or a wardrobe? Were high-quality, robust pieces only consumed by the upper-class? Were there "cheap furniture" options for those less well-to-do?
If an immigrant worker from Sicily is settling in a place like New York City and has now rented their own apartment (bare, unfurnished), how long would it take to have enough money to furnish the place? Where could they go to buy that furniture? What would be the expected quality?
One of the big reasons that surviving furniture pieces are so sturdy is just that the manufacturers didn't have the ability to make them worse.
Two of the major factors that make modern cheap furniture the way it is are particleboard and plastic, the industrial uses of which are considerably more recent than the late nineteenth century. Various types of consolidated wood/pulp materials were being developed at the very end of the century and in the first half of the next; in the period you're talking about, it was only really suitable for molding into decorations to be applied to real wood. It wasn't strong enough to be molded into planks that could take the weight of the furniture itself, or of anything placed on it - so furniture had to be made of solid pieces of wood (even if they were cheaper wood covered with a veneer of nicer material), which hold up better than composites. And bakelite was the first synthetic plastic, developed in 1907, and mainly used at first as a non-conductor in electrical components, leaving sturdier metal brackets, nails, and tacks to be used in furniture construction. Even cheap, shoddily-made furniture that people of the time complained about would generally appear "better" to modern eyes than your basic flat-pack pieces today.
So rather than being made of particleboard, the cheap option was pine (called "deal" in Great Britain at this point, if you come across mentions of deal furniture). It would be made rather or very plainly, with very little ornamentation, not falling much in line with prevailing decorative arts trends - it might not even be varnished. This sort of furniture simply hasn't been of much interest to either museums or antique collectors, so it doesn't have as much visibility, but it is out there.
Another option for saving money was buying secondhand, either through a dealer or at a pawnshop. This could be anything from upscale custom pieces made by an uptown cabinet-maker being resold to cheap pine furniture passing from upper-working-class hands to lower-working-class ones. It's hard to give examples of costs: it would depend so much on the specifics of the furniture, what it was made of and what condition it was in. For instance, in an 1895 court case in New York, a man stated that a suite of furniture he owned (consisting of eight pieces altogether) was bought for about $65, had been devalued to about $50 with use, and had been deemed worth about $10 after a fire. To a poor family, that might be more doable, factoring in a cheap upholstery job with plain cotton, and/or purchasing just a couple of the pieces.
Flat-pack furniture was first manufactured in the mid-nineteenth century. I go into the history of the pioneering Thonet company in an answer here. Thonet and companies in the United States sold what was called “knock-down” furniture made of bentwood. Items including armchairs, rocking chairs, coatracks, and cribs were available at economical prices.
Another innovation that made furniture affordable to the American lower and middle classes was the use of steam-driven machines in manufacturing. Using machines for turning and carving wood meant lower prices and a wider selection of available designs for the end consumer. Mechanization allowed manufacturers to create elaborate designs mimicking more expensive, handmade pieces. Though it was never really cheap, machine-made furniture was affordable enough and of a high enough quality that it could be resold or passed down as an heirloom.
Because of its domination of the market for wooden furniture beginning in the 1870s, Grand Rapids, Michigan, gained the title of “Furniture City”. Furniture in the “Grand Rapids Style” was available for purchase throughout the United States at a range of price points based upon the materials used and the complexity of the construction and in a variety of designs drawn from a mishmash of historical styles that has been described as “synthetic eclecticism”. Consumers knew to look for the name “Grand Rapids” on the label to ensure they were getting the real thing.
This style of furniture was popular well into the twentieth century, though “Grand Rapids” eventually became synonymous with brown furniture for a middle-brow aesthetic. Furniture production slowly moved south (primarily to North Carolina), but Grand Rapids continued to be known for its high-quality office furniture—particularly products made by Herman Miller, Haworth and Steelcase.