I'm not a historian, or even a history or military buff, but I've been thinking about this for a few days now. Military training has varied a LOT throughout history, with different focuses and approaches depending on the military, technological, cultural, and economic situation. When trying to apply these approaches to Dwarf Fortress, this is complicated by a few things.
1. Dwarves aren't humans. They differ from humans with naturally superior strength and toughness. They have different affinities and proclivities than humans usually do. The majority of dwarves seem to live in underground fortresses which enforce a degree of closeness, so dwarven cultures are going to have have a smaller variance than human cultures have in certain aspects. You might expect fortress dwarves to be more cooperative by necessity, and who knows what else. Also, dwarves can enter 'martial trances', which seem to be a weird combination of the flight-or-fight response and a flow state.
2. The world of Dwarf Fortress is a pretty hostile fantasy world. In real life, humans have always fought other humans, with similar physical capabilities. Dwarves are threatened by far more than other dwarven civilizations. The most common threats to a fortress are goblins and elves against which historical military tactics might work well, but fortresses are often attacked by undead, werebeasts, vampires, and megabeasts. A civilization that hasn't gone extinct yet in a world with such varied threats is likely to have developed approaches for dealing with all of the common threats, and clearly you don't find werebeasts the same way you do goblins, or a bronze colossus the same way you do elves.
3. Dwarves in Dwarf Fortress (and really every group) don't actually do battlefield tactics or fight as a unit yet. Fighting in formation was dominant throughout history, and you would expect civilized forces to fight in formation when fighting enemies with similar capabilities. Particularly humans and dwarves seem like they would fight in formation against humans, dwarves, goblins, and elves. Elves might be capable of fighting in formation, but as they are immortal their individual warriors should have individual skill that lessen some of the advantages of fighting in formation, and they also bring war animals that make it less practical to fight in formation. Goblins don't seem likely to fight in formation without the influence of a powerful leader (a goblin warleader rising to prominence could be a neat threat), and as they are also immortal the longest-lived goblins are going to be deadly warriors that have little reason to fight in formation. The lack of actual battlefield tactics right now invalidates a lot of historical training focused on unit tactics.
4. Every fortress starts with only seven dwarves and no military, so fortress-mode training to create a new force isn't going to be the same as the training to refresh an existing military. Unfortunately there isn't a huge amount of writing on the early days of historical military forces, probably because no one thought they were worth writing about until they were already an established force. There is a LOT more writing on modern military training, but I haven't really dug into it.
That all said, there's still useful stuff to be found in how soldiers and warriors trained or were trained throughout history, and there do seem to be some common threads, even ignoring unit tactics entirely. One thing I noticed is that almost every military force of note has been one dedicated to the purpose. Whether by the existence of a hereditary 'warrior class' or a standing, professional army, regular training and some degree of standardized equipment were common.
There are two common aspects that should perhaps inform Dwarf Fortress military training. There are other things, but these are the two most important and uniform things I noticed. Other things like specialization on top of a uniform standard of training are important, but not uniformly true.
Discipline and physical fitness are core to both training and combat performance. The training of professional armies begins with training designed to build fitness and discipline. Members of warrior classes developed physical fitness and discipline alongside and with their education. The recognition of the need for physical fitness dates back to the earliest professional armies and warrior classes. We don't have much writing at all on Tiglath-Pileser III's standing army, or on the training of the ancient Macedonian army, the first well-documented standing army. However, we know that the Spartans who centered their society on their warrior class put heavy emphasis on physical fitness as a part of early education. We have much better documentation of the training of the Roman Army. Before roman recruits trained with weapons at all, their training focused on discipline and physical fitness while simultaneously training important soldierly skills like marching and digging. The same is true of military training today. Even after training is complete, effective military forces attempt to maintain physical fitness. Roman soldiers trained every day to stay fit. Modern military forces do the same. Discipline is as important as fitness. Both warrior cultures and professional militaries place heavy importance on obeying orders, courage in battle, and often on self-discipline off the field of battle. Both ancient and modern forces use drills to teach soldiers how to behave in combat, to make combat regular so that when it happens for real they won't freeze up and will take the right action. After training, drills, inspections, labor, and field exercises help to maintain individual and unit discipline much as regular exercise maintains fitness.
Fresh recruits are trained by the experienced. A new recruit is assumed to have zero applicable skills. After developing some degree of fitness and discipline, they begin training in the various skills a soldier is expected to have. First a recruit is trained in the basics of a skill. Once the basics are developed, the skill itself can be trained. Usually they are taught by a specialist with training in the skill, with a training program designed to improve their skills. Many skills cannot be practically developed from scratch by independent practice. Those that can still generally require some sort of training program, so the person practicing can measure where and how they can improve. A great example is firearms marksmanship. It is possible to go from no experience with a gun to a great shot, but the initial stage of learning how to use the gun and the basics of aiming and shooting goes much faster with someone to instruct you in the process. Even after you have some degree of proficiency, simply shooting at a target won't make you a better marksman. Practice can help you maintain your level of skill, but to improve you need to identify where and why you are failing and seek to improve that aspect. The same is true of many skills, so military training tries to provides the basics and then puts the recruit on a training program designed to develop the skill to a target level. Of course, not every skill is so easily measured as marksmanship, and not every skill can be usefully trained solo. Hand-to-hand combat both in the military and in civilian life is trained by a combination of guided drills, sparring, and direct instruction. Drills help teach people the movements and build speed, accuracy, and power, but actual fighting can only be taught by experience. Even so, the experience will be more useful if there is an instructor to provide perspective on what you did right or wrong.
It's also important to note that learning continues after basic training. Historically, 'green' units comprised of all or mostly fresh recruits have performed much worse than units that were mostly veterans. New recruits whether enlisted or officers can learn from their peers, superiors, and subordinates.
How to apply this to DF fortress mode training? I don't know, but I'm seeing two things the current system lacks. One thing the current fortress-mode system lacks is a way to focus on physical training; right now, most people who are going for a highly-trained military pre-train their soldiers with mining or pump operating. The second thing we're lacking is a way for dwarves to be trained in skills by dwarves experienced in those skills. This could be achieved by the addition of more dwarf behaviours, or by the addition of features like seconding or per-barracks instructors. Adding more micro is undesirable but currently training recruits with an experienced soldier is pretty clunky, so the addition of new features will probably be a reduction in micro.