The "I'm a..." format of questions unfortunately turns off the majority of flaired users. If you don't get a response, try submitting again framing the question normally, like your follow-ups.

The foundation of Chinese education during the Late Imperial period was the civil service examination. The state would hold competitive examinations of students at the district, county, prefecture, provincial, and imperial levels, which would cover the Confucian canon, and presumed knowledge of all the arts and skills of a Chinese gentleman (ritual, archery, Go, calligraphy, painting, and so on). Education in imperial China revolved around these examinations, and for good reason. The government issued degrees to a small number of scholars at each level (in Hunan, 3,600 students were admitted to the provincial examination; 45 degrees were issued; I believe Zeng Guofan's father took the provincial examination seventeen times before passing), and government officials were recruited from the ranks of degree holders. However, even the lowest degrees had some perks; Shengyuan degree holders were exempt from corvee labor, had protection against corporal punishment, and the highest grade of Shengyuan got a stipend from the government so they could study full time. I shamelessly screenshotted a flowchart Dr. Man-Cheong included in her book, The Class of 1761, which you should definitely read if you want a full picture.

https://41.media.tumblr.com/f7f7487b57bd5fe4d97147bb42c8c8e7/tumblr_inline_nly7z2vDGp1rf4l09_500.png

https://40.media.tumblr.com/15f003b7014f5b9f5f4edea876361128/tumblr_inline_nly7zlLqGH1rf4l09_500.png

As you can see, keeping your Shengyuan degree takes a lot of work; you have to take four seasonal examinations and one annual examination until you pass the provincial examination, which often took decades, since the pass rate was often around one in seventy (and even then, the Qianlong emperor lamented that there was still an oversupply of qualified degree holders for the top level positions). To circumvent this, there arose a class of students called Gongsheng, or tributary students; they studied at the capital in the Taixue, or university, and were exempt from the continual reexamination of the regular Shengyuan students.

There were five kinds of Tribute Students; Excellent Tribute Students (You Gongsheng) were selected by the provincial director of education, originally as a form of patronage between teacher and student, but the procedure took an additional exam, as well as an imperial interview, thus undermining the local patronage aspect.. Honorable Mention Tribute Studets, or Fuban Gongsheng, were those who had failed the provincial examination, but had performed well enough that they were exempted from the perpetual reexamination of the rank and file Shengyuan. Ba Gongsheng, the most common type, had also distinguished themselves, though not many felt highly of the privilege, having failed to even mention it on their applications. Sui Gongsheng were exempted from the continual reexamination as a show of Imperial beneficence once they'd failed the provincial examination ten times, and En Gongsheng received the privilege in celebration of Imperial birthdays, new conquests, and other state mandated celebrations.

Fortunately for the wealthy, it was possible to straight up pay your way out of this labyrinth of degrees and studentships; Jiansheng, or Imperial University Student status could be purchased for about a hundred taels of silver, and it exempted you from having to take the district->county->prefecture-> Shengyuan examinations. Those holding this degree could sit at the provincial examinations in Shuntian, Beijing directly. After one achieves a provincial degree, the pace of examinations slows considerably, allowing students to find time to work in lower government offices and study for the real examinations in the capital and the palace itself.

So that's the system; in terms of content, it's very much focused on classic literature. There's a lot of rote memorization involved; when answering the questions, one would be expected to quote from the Confucian classics and the extensive relevant commentaries at length, and a single misplaced stroke could be grounds for failure. The Confucian classics cover a wide variety of subjects, such as history, arts, divination, ethics, ritual, and later additions to the Confucian interpretive framework during the Song and Ming dynasties expanded the subject matter of scholarly inquiry to metaphysics and (proto-)archaeology, as well as Legalist style uniform law codes. I don't know if it was ever included on the examinations, but Chinese mathematics received a shot in the arm with the introduction of geometric proofs (along with astronomy and cannon casting) by the Jesuits. In preparation for the exams, one would often hire a qualified tutor without other employment, and commonly serve as a tutor or teacher oneself upon reaching the appropriate level; wealthy students could also purchase winning answers from previous examinations to see what the examiners responded favorably to. Still, even those who failed to make any progress in the examinations received plenty of social benefits; one who had taken the examinations became an important man in his community. Simply being literate put one in the elite, and having knowledge of law and ritual put them in a good position to resolve minor disputes and lead local ceremonies, which lent them great prestige.