Note: I'm interested in any historic militaries, not just Rome's or China's.
First, my understanding of bootcamp is heavily colored by depictions of bootcamp in popular culture (i.e. in Full Metal Jacket or Band of Brothers). While I've heard these are fairly accurate, I do wonder how they differ from the reality, both of training for WWI/WWII/Vietnam and from contemporary training.
That said, some more specific questions:
How did physical training change over time? Were the obstacle courses seen in portrayals of US Army bootcamp present in historic training regimes, or are they a recent addition? What about running?
What about the psychological aspects of boot camp? Did historic training involve the same attempts to psychologically break down recruits? Where different techniques used by different cultures?
How were recruits chosen? Were they draftees, volunteers, or something else?
What about instructors? Where they military? Did individual units train their own troops, or were national training standards common?
How long was training? Was it a relatively short program intended for adults (like modern bootcamps), or something that occupied a soldier's entire childhood (like ancient Sparta)?
China has a long history of warfare, and thus it should not be surprising that boot camps, or rather military training camps have existed from a long while.
The first recorded mention of military training is in the Book of Rites. There it indicates that six form of arts were compulsory among all students during the Western Zhou dynasty (1046 BC - 256 BC) - rites, music, shooting, riding, reading, and counting. The military training provided will be allocated according to the appropriate season, where in the first half of the season was spent learning and training with regular screenings, while the second half of the season would be more focused on hunts and mock battles for practice. In Spring, the training would stress on basic movements and formations - standing, crawling, attacking, retreating, speeding up, slowing down, gather together, and divide into groups. Summer would concentrate more on the surroundings - how to proceed across terrain, and how to survive on the terrain when on the move. Autumn would bring more formation training, as well as weapons training and actual combat. Winter would end with an overall screening of the army, as well as application of what the students have learnt throughout the year.
Shooting lessons were especially important, because the student's future depends on it. The chapter on the art of shooting within the Book of Rites states that the king decides on which courtiers to appoint to higher position based on their skill in shooting, and that those who perform well may be promoted while those who did badly would be reprimanded and may even be demoted.
After having been trained, the students will be tested in five key areas - to ensure that the arrows pierce through the target board, to be able to shoot consecutive arrows quickly without hesitation, to shoot arrows that fly straight and true to ensure a quick and fatal hit, to maintain proper positioning throughout and be extremely accurate, to shoot four arrows which would land to shape like a square. The difficulty of these tests demonstrate that the training must have been quite intense, to expect students all to maintain such a standard.
Likewise, the riding art would test students at the skills in riding chariots. This would involve something like an obstacle course, with students requiring to navigate pass objects, across streams, avoiding other chariots and making use of the surroundings to their advantage. This was where mock battles would be important, to better hone the students' skills on the chariot.
The progression for training can be seen from Six Strategies for War, where Jiang Shang explains to Ji Fa, king of Zhou - 'firstly it starts with one-to-one training, when he learns proceed to training a group of ten, when they learn then train a group of a hundred, when they learn then train a group of a thousand, when they learn then train a group of ten thousand, when they learn then train the entire army altogether.' I'd advise you to read the book itself for specifics like what kind of soldiers are suited to what roles, how each section of the army should be utilized, etc.
Onto the spring autumn period, military training became even more important since the ability of the army would decide the state's survival. In Records of the Grand Historian we have the famous account of how Sun Wu managed to even whip the palace concubines into a workable military troop - starting from basic movements like turn left and turn right, but drilling them to such an extent that they do so in unison and without hesitation. With the threat of death behind them, Sun Wu declared that the all female troop would be able to brave all elements to fulfill their duty.
Also gaining in popularity during this period of time was martial dance 武舞. This form of dance would usually include the use of weapons and occasionally would involve two participants alternately attacking or defending to better show off their skills. Much like the martial arts display in modern times, the martial dance required plenty of training, as well as concentration and physical ability, to perform well.
However, military training soon came to a halt during the Qin dynasty. In order to discourage rebellions and revolts, the emperor demilitarized the empire, especially among the common folk, to ensure that weapons were out of reach. Even knives for cooking became limited in number, and those who wished to train had to do so on the sly.
As time passed, training for the common people dwindled, save for those who could afford to go to the more exclusive private schools who maintained the standards from the Zhou dynasty. The Sui dynasty was especially against military training for the masses, instead practicing the idea of 'putting down warfare to learn the literary arts' 偃武修文. Schools split up the tests for martial and literary arts, making shooting and riding no longer compulsory for students who wished to just concentrate on book studies. Since the courtiers were no longer rewarded for exceptionally good martial skills, most turned to just being literary scholars and gave up military training altogether.
An interesting fact is that football was fairly often used as a form of recreation, but also as training for the army. Han general Huo Qubing often kept his troops fit this way, so as to have them in top condition when battling against the Xiongnu.
In modern China, military training has since once again become part of the student curriculum for secondary and university students. So in a way, China has just come full circle.