One particularly notable trainer (though in terms of drill rather than teaching strategy) was the one who rose from a low birth to become Emperor: Maximinus Thrax. Recorded in various ancient historians but see for instance Herodian discussing him here: http://www.livius.org/sources/content/herodian-s-roman-history/herodian-6.8/

[6.8.2] Because of his military experience, which I have noted above, Alexander put Maximinus in charge of training recruits for the entire army; his task was to instruct them in military duties and prepare them for service in war. By carrying out his assignments thoroughly and diligently, Maximinus won the affection of the soldiers. He not only taught them their duties; he also demonstrated personally to each man what he was to do. As a result, the recruits imitated his manliness and were both his pupils and his admirers.

[6.8.3] He won their devotion by giving them all kinds of gifts and rewards. Consequently, the recruits, who included an especially large number of Pannonians, praised the masculinity of Maximinus and despised Alexander as a mother's boy. Their contempt for the emperor was increased by the fact that the empire was being managed by a woman's authority and a woman's judgment, and by the fact that Alexander had directed the campaigns carelessly and timidly. They reminded each other of the defeats in the East which had resulted from the emperor's negligence and of his failure to do anything courageous or vigorous when he faced the Germans.

[6.8.4] The soldiers were therefore ready for a change of emperors. They had additional reasons for discontent: they considered the current reign burdensome because of its long duration; they thought it profitless for them now that all rivalry had been eliminated; and they hoped that the reign which they intended to institute would be advantageous to them and that the empire would be much coveted and highly valued by a man who received it unexpectedly. They plotted now to kill Alexander and proclaim Maximinus emperor and Augustus, since he was their fellow soldier and messmate and seemed, because of his experience and courage, to be the right man to take charge of the present war.

[6.8.5] [February 235] They therefore assembled on the drill field for their regular training; when Maximinus took his position before them, either unaware of what was happening or having secretly made prior preparations for the event, the soldiers robed him in the imperial purple and proclaimed him emperor.

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I'm not sure if they had anything like a military academy, rather than relying on young nobles serving alongside experienced generals and learning the ropes. Would be really interested in answers on that side!