If sent to training, what is it like? Is the path anything similar to current army trainings nowadays?
Training was different in different periods for different units of different origins, but as you have specified that you are a Roman/Italian male in the 1st century BC, you have made this question much easier to answer. So thanks!
Firstly, there were no standard physical training institutions in this period: there was no Parris Island or Fort Jackson (or Kapooka if you're Australian like me). A new recruit trained with his legion, at the legion's camp, and this camp could move around as part of a recruit's training was marching around and also learning to build the camp! So the location of the training in this context would almost certainly be in Italy, as all Rome's problems basically occurred around Italy in this period: endless civil wars and the Gallic wars, with some prominent exceptions such as the catastrophe at Carrhae.
Drill and "close order formation" did not dominate Roman training like you might think. They never really learned to march in step, and their battle formations were relatively loose, though still cohesive and highly organised. Whereas drill discipline was more important to, for instance, Greeks and their precisely deployed hoplites, the Romans were more interested in developing the personal qualities that they attributed to a good soldier: virtus (courage) and disciplina. The former was believed by the Romans to be a product of one's natural character, but they also understood training and skill to be essential components of a soldier's confidence. The Romans certainly did weapons training with sword and/or pilum, perhaps daily. As I mentioned they also did a lot of marching with full equipment to harden themselves. Individual combat skill was perhaps the most prized quality of legionaries in this period.
As opposed to virtus which could only be enhanced by training and was 'summoned' by a good general, disciplina was something that the Romans understood could be developed through training and hard work. The Roman elite had a moral fear of cowardice and luxury, and disciplina was the opposite of this. It could not be developed in a 3 month basic course, but was a product of constant labour. Soldiers continued to train throughout their careers, and as experience was the best teacher, even a well trained army like Marius's or Caesar's had to be blooded carefully until they were truly ready for an all-costs pitched battle. Armies that continued to serve and continued to fight pitched battles became incredibly efficient, as did Caesar's. This wasn't just their combat experience, but they also never stopped training, developing and maintaining their disciplina.
As civil war dominated Roman warfare in the 1st century BC, as a young Roman man "signing up" (you would actually be conscripted, in this context of civil strife you would be unlikely to volunteer for service unless you were a struggling veteran) would be scooped up by whichever general you now work for and set to work by his centurions. You would be trained daily in the use of weapons, you would do a lot of marching, you would do a lot of physical labour and work on camps, and eventually you would probably participate in unit exercises, hopefully before you had to fight a battle. This might take a few months, but as evidenced by Marius' extensive preparations spanning years before Aquae Sextiae, you could possibly never have too much training. That said, huge numbers of Italian/Roman military age men during this period had military experience, but as things got worse people got more reluctant to serve. Pompey's reputation made it easy for him to call a large army back "to the standards" when his war with Caesar started, but as the chaos of the 1st century BC wore on, coercive measures became more and more important in motivating soldiers to fight. Decimation became common again during this period after falling out of fashion in the 2nd century, for instance.
A nice account (written c. 100 AD however) of Roman training exists in Plutarch's Life of Marius. See 13, 14.1-3, 15-19 for Marius' detailed inculcation of confidence into his troops. Note that once his soldiers had properly developed their virtus, they were so aggressive that even Marius could not control them at times. Caesar also manages to barely restrain the aggression of his troops in Gaul at times, most prominently at Gergovia where they allegedly attack without orders and suffer a defeat. It's also not the only time Caesar has this problem.* Contrast this with Caesar's less-confident army in book one, which nearly mutinies to avoid fighting the Germans, who have a formidable reputation (Bellum Gallicum 1.39-40). The importance of experience here is highlighted next to training.
Overall, you the inexperienced young Roman man who has "signed up" (been conscripted) in this period, will go straight to training. But you'll be training with the army, which may be off to battle in the next few weeks or months. You'll live in the camp, you'll do a lot of marching, swinging swords and throwing javelins, you'll be introduced to military discipline, and you'll do a lot of manual labour. You don't need to be sent to the legion, the legion most likely came to you!
Sources:
Caesar, Bellum Gallicum.
Plutarch, Life of Marius.
Phang, Sara E. Roman Military Service: Ideologies of Discipline in the Late Republic and Early Principate, New York, Cambridge University Press, 2008
You went straight to the legion (and this goes for both the Republic and the Empire). There was no formalized "basic training" or "boot camp." Instead, training was conducted on the march, or whenever there was time. Unfortunately the ancient sources are not very forthcoming about how training worked; our best source for this is Vegetius, writing in the late 4th century CE, when many aspects of the legionary life had changed drastically from the Republic and early Imperial periods. Training regimens seem to have been left up to the individual commanders, with some demanding more rigor and some less. In the Imperial period we hear about campidoctores from inscriptions, and they had some kind of training role. In general, training once or twice per day, either in melee or formation training, seems about right, and Vegetius says something similar (once per day for the veterans, twice for raw recruits, and formation drills two or three times a month; Mil. 2.23; 27). Josephus (BJ 3.70-74) quips that Roman legions' training is a bloodless battle, and their battles bloody maneuvers, suggesting that at least in his limited experience, formation drilling or large-scale war games were possible.
There exists a trope in Latin literature of the capable Roman general teaching his troops individually. So Plutarch (CM 20.4) tells us that Cato the Elder taught his son to throw the pilum, wear armor properly, ride a horse, swim, and win in a boxing match. He has similar stories of both Marius and Pompey training with soldiers on the Campus Martius, teaching them to fight (Plut. Mar. 34; Pomp. 64). Pliny talks about the emperor Trajan doing the same, and we hear about Hadrian demonstrating horse skills. This theme is wrapped up in Roman ideas of virtus, manly battlefield prowess, and it is hard to cut through it and get to the specifics of actual training. Surely Pompey did not line his legions up and spar each man in turn. 20th century scholars usually argued that the centurions shouldered the brunt of the training "in the field," just as the NCOs do in modern armies; recently some have pushed back against this argument.
After a recruit was selected, he underwent a very brief evaluation called the tirocinium. I confess I know almost nothing about this, and it is possible that some source somewhere has some information (hopefully besides Vegetius). There were different kinds of tirocinia, the most talked-about being the tirocinium fori, a sort of apprenticeship for young politicians. In the army it probably had an evaluative role, like checking eyesight, physical fitness, et cetera.