Sorry for the delayed response. I've been at work the past two days and haven't really had the time to crack a book and do some research.

As to your question, the sources I have don't really speak to a specific time during the professional Roman army like Caesar's conquest of Gaul, but does provide a broad idea of what the recruitment/training process was like for your average soldier during the Principate.

The process for recruiting was universal across the empire and consisted of a multi-stage process. The first thing potential recruits were subjected to was the probatio, a kind of general inspection. Each recruit's legal status was verified at this point, for only Roman citizens were permitted to join the legions. Slaves especially were barred from any sort of service whatsoever, and if you were a slave pretending to be a free man joining the army, your punishment would be severe. The importance of this stage of the probatio cannot be overstated. Citizenship meant everything to Romans. As an example, in the year 92 AD, an optio serving in Legio III Cyrenaica was accused of not being a Roman citizen and therefore faced dismissal from the army at the very least for his transgression. It took three witnesses, two of them legionaries from other centuries, to prove his status as a bona fide citizen.

Once a recruit's legal status had been established, the second half of the probatio took place. This consisted of a simple medical examination to determine whether or not a man could perform the duties of a soldier. One papyrus records the discharge of a soldier due to weak eyesight, presumably caused by a cataract. Something like poor vision, lameness in a limb, or other medial maladies usually lead to automatic disqualification. This part of the process seems pretty straightforward and would be similar to a medical examination today.

Recruits who passed the probatio were accepted for service and would be sent to their respective units. At this time the recruits would be given the signaculum, a lead tablet worn around the neck that would identify the soldier, similar to a dog tag. Soldiers at this time would take the military oath, or sacramentum, to the emperor. Once the recruits arrived at their legion, basic training could begin.

Soldiers were first drilled and trained in physical fitness and discipline. Close order drills and marches were used to teach recruits to stay in formation. The marching not only trained soldiers to work together but also improved their stamina. According to Vegetius, roman soldiers were expected to march twenty roman miles in five hours, or 24 miles at forced march. Much like we train soldiers today, as recruits advanced in training, much of their fitness routines were performed with uniform and backpack. Beyond mere marching and drills, soldiers were trained in the effective use of the gladius. The system the Roman legions used to train swordsmanship borrowed heavily from gladiatorial training. A six foot tall post would be erected and a trainee would aim blows at the object with a wooden sword and wicker shield. These posts would also be used as targets for the throwing of pila.

Mock battles would eventually be fought between legions in full gear to prepare recruits for the din of battle. Soldiers would either use their wooden weapons or sheath an actual gladius in leather to prevent it from harming fellow legionaries. According to Adrian Goldsworthy, in his book The Complete Roman Army, basic training to bring a recruit up to the level of an actual legionary took several months. The training, of course, didn't stop there. Commanders were expected to keep their men well drilled and at constant battle readiness.

Once a recruit fell in with a legion, that legion formed the basis for their identity. Legions were fiercely proud of their accomplishments, and conversely held other legions in contempt. With each having unique dress, unique traditions, and therefore a unique identity, strong rivalries were established. Each legion had its own standards and symbols they would emblazon on their shields. Legion names often expressed martial virtues or regions in which they'd served with distinction, something for which soldiers even decades removed from the event that granted the legion its name, could feel proud. Names such as Legio VI Ferrata fidelis constans (Ironsides, faithful and constant, founded by Caesar) or Legio XXI Rapax ( "Grasping" or greedy for victory) would, in theory, instill a sense of pride in every soldier in the legion. Soldiers' tombstones, even when the inscription was short, would often display the legion in which the deceased had served.

I don't have much more on traditions side of things. If you are interested in the Roman army I suggest picking up Adrian Goldsworthy's book The Complete Roman Army. It is a wealth of knowledge.