I want to note from the outset that my answer, due to gaps in my own knowledge as well as a general tendency in the scholarship, is pretty focused on the Northeast, and is entirely focused on the White working class. There are important parts of this story that are about Southern Whites as well as Blacks, both before and after the Civil War. I hope someone can fill in that story.

The nineteenth century saw the transformation of the American economy from, as you are aware, a largely agricultural focus to a largely industrial one. For a native-born White man at the dawn of industrialization (the end of the eighteenth or the beginning of the nineteenth century), the most likely occupation was farm labor, usually on a family-owned farm in the Northeast, Old Northwest, or the South (particularly the Upper South). Gradually, these lands became less viable, due to soil exhaustion and partition of family farms into smaller and smaller chunks as a result of generations of inheritance. For the young men growing up on these farms, there were a couple main options: One was emigration, going either alone or with a young family to the newly-acquired territory in the West (this includes the Louisiana Purchase and all the land acquired from Mexico, plus the sections of the Oregon territory that ended up in the U.S. after the treaty with Britain). The other was to move to the city and join attempt to join the burgeoning middle class by acquiring work as a clerk in a mercantile house, eventually amassing enough money to start one's own business or otherwise join the ranks of the professionals (studying the law and, later in the century, medicine were also options for this route). However, this isn't really the "working class" in the industrial sense.

The working class is a creation of industrial capitalism. Before the widespread introduction of that mode of social organization and production, the urban producing classes were artisans. Organized in a manner very similar to the earliest medieval guilds, each trade would have had its own organization of master craftsmen, journeymen, and apprenticeship, and would have governed its own norms. This includes things like shoemakers, coopers, tailors, etc. Traditional norms would have governed their working lives, including a principal of independence for both shop owners and employees. Importantly, the "management" (ie the master artisans) would have closely identified with the workers. In other words, hierarchies were relatively flat, and there was a strong sense of a group of brother tradesmen working together. This included the right to stop working when one felt like it, to drink on the job, and to otherwise organize the day how one saw fit. What happened to this way of life? Industrial capitalism. As machinery increased the efficiency of production of basic goods like shoes, clothes, barrels, etc., it also decreased the skill required to produce them. This proved disastrous to the working class, who found themselves suddenly faced with an influx of immigrants willing to work for much lower wages operating the new machinery, causing severe downward wage pressure. Additionally, the capitalists took this opportunity to begin to assert greater control over the mode of work, requiring longer working hours with less freedom (including the banning of drinking on the job, or of the practice of knocking off for an impromptu holiday). The trade guilds fought back as best they could, forming the kernel of the American labor union movement.

As for what jobs they did, perhaps an economic historian will have actual numbers, but throughout the nineteenth century most jobs would still have been in farming. Resource extraction (mining, fishing, oil, etc) would have been big ones, as would manufacturing. You also have various labor jobs, like teamsters and longshoremen, supporting the import and export of all this raw material and finished goods. As I mentioned, for the emergent middle class you would have had the professions, like law, as well as accounting and the aspiration to ownership and upward mobility. Finally, there were a class of very low-paying, undesirable jobs like rag- and bone-picker, men who would walk the streets going through rubbish looking for rags (for papermakers) and bones (for soap-making as well as knife handles and the like). These were a group of true underclass, living in absolute penury and making the barest of livings.

I hope this brings some clarity. Immigration levels were very high, and for the men who did not make it to the farms of the Midwest or into the middle and managerial classes, they would have been caught up in the same forces of industrial capitalism that swallowed up all the immigrants, though their relatively higher levels of social and actual capital would have actually helped. But do keep in mind that most labor was still agricultural throughout the nineteenth century. It was a process of urbanization and industrialization, rather than all-at-once.

Sources:

Wilentz, Chants Democratic

Blight, The Moneyed Metropolis

Trachtenberg, The Incorporation of America