Oddly, this describes the very early life of Caligula. As a toddler, Caligula joined his father while he was campaigning in Germany and would dress in a miniature solider's uniform. So he was called Caligula because of the little boots that he wore (the root word is caliga). He accompanied his father until his father was killed in 19 AD/CE.

However, it may be an isolated incident. I looked into the early life of Tacitus but I haven't read enough of him to say whether he was a part of Agricola's campaigns. It might be up to the father and how important he was/where he was stationed to allow his family to join.

Just a clarification. This would be the realm of the elite families, both patrician and plebeian. One's father would not be a 'general' in our modern sense. There was no line (in our age, a massive concrete reinforced wall with towers and MG's) between the civil and military - a magistrate would be expected to deliver judgments in the forum as well as leading men into battle.

So if you were the son of a Roman elite with you'd almost certainly have had some military experience by the time you were a young adult. As family members or friends climbed the cursus honorem of offices, they'd get military commands (sometimes during their year in office, but more often after their year when they were assigned to their province) as either a propraetor or proconsul. These magistrates would bring along their own council of advisors: family friends, younger sons/relatives. The elder ones would provide advice while the younger would watch and learn.

By the late Republic, magistrates were constantly seeking excuses to fight battles and thus gain glory that could be cashed in on their return for a triumph (or lesser ovation) which would boost their chances for the next office up. So as a teen on the cusp of manhood you'd probably have plenty of chances to join a magistrates' staff and see campaigns and battle.

Under the Principate, the elites no longer had the combined civil/military career paths as the emperors had no desire to see successful senators rise up and challenge them. There your appointment to command a legion came only if you had the favour of the emperor and the duration was an average of 3yrs. Between that appointment would see a variety of purely civil positions culminating in the appointment as governor of a province.

In the Principate there was a formal path for the sons of the elite. Most would be being appointed a tribunus laticlavius (one of 6 in a legion) where they'd serve for a minimum of a year. At 24 they'd be enrolled in the senate. From then on he'd do multiple magistracies (with less power and no military side) and climb the greasy pole. At 30 he'd be eligible to be a legatus legionis. It seems that the tribunus laticlavius often served in provinces governed by family or close friends so possibly they were specifically requested.

In the pre-Marian reforms period of the Roman Republic, there was no standing army per se. Most land owning citizens would take up arms during war (to defend their lands and to gain war booty - soldiers were not paid otherwise), and the Italian allies would send auxiliaries. In fact, you even had to buy or make your own armour. Regardless of the status of your father, you were expected to take up arms as a Roman citizen during your teens (which is when Roman boys came into manhood).

All patrician men would definitely have taken up combat, as ten years of military service for the Republic was the only way to be elected into the senate. After political rights apart from the post of Plebian tribune opened up to the Plebian citizens of Rome, this requirement continued to be imposed upon them.

So yes, you would be expected to go on campaign. But not with him necessarily. Not until the Marian reforms, when generals started to become the main source of power in the Republic as opposed to the senate.