I can't speak to all companies, but I can to some extent for the East India Company. The big names you'd want to check out here are H.V Bowen, in particular his Business of Empire. Another useful source is Bernard Cohn and his article The Recruitment and Training of British Civil Servants in India, 1600-1860 in "An Anthropologist Among the Historians." Both of these engage with what it was like to join and work for the East India Company as an ordinary servant.

With the EIC, you typically joined at a young age. It varied over the years, but broadly it lasted from about 15-19. The East India Company had a core of employees known as the Covenanted Servants, because they took an oath of loyalty to the EIC. The only way to do this was to be nominated by a Director of the Company. Over the course of the 19th century, additional layers appeared, namely an initial training institution called Haileybury college (after a similar attempt in the early years of the 19th century to establish training institutions in Kolkata) to train these servants as the EIC was transformed from a mercantile company into an administrative entity ruling an empire.

Company Servants were, nominally, merchants. Their job however was often mercantile and administrative. They would work as merchants for the Company, while also eventually running company institutions such as their factories in places such as India, Singapore, China, etc. (Note, Factories are not like modern entities. EIC Factories were basically warehouses and exchange centers where inventory was stocked, and records were maintained). Company Servants ran their own private businesses on the side, mostly mercantile, but some also conducting professions such as becoming lawyers in the various EIC territories. These private practices enjoyed various levels of unofficial approval, though their private trade wasn't always condoned. Much of the history of the EIC was infact its servants invariably prioritizing their private trade enterprises over the Company's, building their fortunes rather than the Company's.

A young Company servant would originally start out in the equivalent of a probate employee, or like an apprentice. They would perform relatively menial tasks, such as the manual compilation of written records. This is why the lowest rung of employees were literally called writers. To this day, that name has survived in Kolkata in the form of "Writer's Building" the original center of the state government machinery. Essentially these writers would copy, maintain records for dispatch to other parts of the colony, satisfy requests for documentation, etc.

Eventually they would rise through the Company hierarchy, rising into various mercantile and administrative positions. Networking was key to your rise in the company, and most servants would hitch themselves to the coattails of more senior servants. This would result in the creation of factions and groups within the Company, many often with ties straight to the Company elite in their Court of Directors. These factions in turn reinforced themselves by controlling the entry of new employees.

A Company servant could, quite easily, look to spend the better part of 40-60 years working for the Company. Plenty would die in service, and while some would rise no further, one could, for most of its history, keep rising until one ended up the equivalent of a colonial governor at the end of it. Even company directors. Though lateral entry did exist, old school company servants recruited at a young age were the ones who had the best chances of continuously rising through the ranks.

For much of the English "middle classes" of urban and suburban professionals, mercantile groups, etc (comparable somewhat to the petty bourgeoise I suppose) the various Companies often represented the best way of gaining social and economic standing in British society. The EIC in particular was dominated by English and Scottish middle class entrants, and by the late 18th century a career in the EIC could be spectacularly rewarding. One might gain enough wealth to rise several rungs in English society, retiring as minor landed gentry, and capable of raising children as gentlemen. But some of the most successful Company Servants such as Robert Clive or Warren Hastings could make so much money that they could quite literally buy out baronetcies and end up as English Aristocracy. This ability to make obscene amounts of wealth eventually became something of a scandal in England, and would contribute to the dismantling of the various companies. Known most infamous as Nabobs (a corruption of the word Nawab, meaning an Indian prince) the scandal of Company Wealth and corruption became something the likes of Edmund Burke attacked ferociously (See for instance Nicholas Dirks, Scandal of Empire, and P.J Marshall's Impeachment of Warren Hastings)

For what its worth, the experience of working in the Company isn't quite analogous to working in a modern corporation. A far better analogue is the Civil Service, which in many ways were structured on the hierarchies of these companies. The British Home Civil Service in particular quite explicitly owes its structural roots to the EIC and the Indian Civil Service it created. More than just the first multi-national corporations, the various Companies were also among the first organized bureaucracies.