The processes for learning Gaulish and Greek would have been quite different. This difference stems from the relative status of the two languages: in Rome, starting in the 2nd Century BC, Greek was a language of the educated elite, while other languages were simply not viewed with the same level of prestige (in fact, some believed that Latin was a descendent or a bastardized version of Greek). As such, we have a better idea of what education would be for Greek or Latin than we do for other languages. I will address the ways in which a Roman citizen would learn Latin and Greek, and then talk (briefly) about how one might go about learning another language, and in what circumstances one might have that inclination.

(It should be noted here that I have taken "Roman citizen" from the question to mean "a native Latin speaker from Italy", which is not precisely the definition; further, this answer is chronologically accurate for roughly the first century BC and the first century AD).

Learning Greek and Latin

The relationship between Latin and Greek in the Roman eyes was quite complicated. On the one hand, many educated Romans spoke Greek, and viewed Greek culture as superior to Roman in many ways. On the other, this self-imposed inferiority led them to be defensive around certain uses of Greek, especially in formal contexts, even when able to speak it. Lucullus, a politician and a writer of history from the first century BC, is said by Cicero to intentionally make errors in his Greek so as to avoid a perception that he was too Hellenized. Cicero himself was criticized for addressing the senate in Syracuse in Greek--his critics felt that he demeaned the status of Rome by addressing a foreign nation in a language other than Latin. (This conflict of which language to speak in is still seen to the present day in international relations--diplomats or politicians often use translators even when fluent in the language of whoever they're talking to.) However, Greek was spoken (untranslated) by emissaries from Greek city-states in the Roman senate, indicating a generally high level of understanding of the language among the senators.

While the extent to which fluency in Greek extended is debated (some scholars believe that the amount of anecdotes we have about Roman elites speaking Greek is indicative of the rarity or impressiveness of that feat, and not of the widespread nature of it), it is certain that higher literary education involved Greek. After the 2nd Century BC, a Roman student would have gone to school from the age of around 6. The very poor never attended school, and the very wealthy sometimes took classes at their home, from private tutors, for the first few years. Those exceptions aside, the vast majority of Roman children attended school from ages 6 to 12. Wealthy students would often attend school until 18, and this was divided further into two three-year blocks: from 12-15, a student would attend the school of the grammaticus, where one learned, unsurprisingly, proper grammar; from 15-18 the student would attend the rhetor, and learn the finer points of oratory. Until around 25 BC, midway through the so-called "golden age" of Latin literature, the literature taught in these schools was primarily in Greek, and frequently teachers were Greeks as well.

In previous years, before when the system described above came into place, Roman children would have been educated at home, by their parents. This early education--like the later education--was unabashedly practical. Romans had no use for theoretical medicines, sciences or mathematics, and only cared about these subjects as they applied to the real world--which they applied brilliantly in, e.g., architecture. Similarly, while the Romans exercised, they did so to remain fit for military service, and not to run races, or wrestle, or throw the discus, or compete in the Olympics, like the Greeks did. The early education was designed to make a child into a man of a certain type of character. He learned, in the words of scholar J.J. Eyre, “A way of life, rather than a body of knowledge.” The later Roman education, described above, was also a practical education, though the aim had shifted from a certain character to oratorical skill. The pupil who had completed this educational track would be an excellent orator.

There was certainly Latin-Greek bilingualism among members of society who were not wealthy or especially well-educated as well. There is a surviving corpus of inscriptions that were clearly made by people who were literate in both Greek and Latin, but who had not been trained in a school of rhetoric. These people were either native Greek speakers, who were slaves living in Italy, or Roman officials living in Greece, or in some other category—simply put, Latin-Greek bilingualism was seen across all social classes with what the evidence suggests is reasonable frequency.

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Learning a Different Language

The path to learning a non-Greek language would have been very different. Certainly some people who lived in areas subjugated by the Romans would learn Latin, and thus become bilingual, but that was not as ubiquitous as would maybe be expected. Take, for example, Germanic: after the Germanic-speaking peoples were incorporated into the Roman empire, there would have been surprisingly little incentive (and no government mandate or requirement) for a Germanic speaker to learn Latin. Even in the military, there were monolingual Germanic speakers serving in auxiliary units which were commanded by Germanic-speaking officers. That said, the army was a path to bilingualism, and for an officer in a Germanic-speaking unit it was certainly a requirement.

Two interesting things should be noted about Germanic-Latin bilingualism which are useful examples for bilingualism in the Roman Empire more broadly. The first is that almost always the Germanic-Latin bilingual was a native Germanic speaker, who had learned Latin later. (This is certainly true for other languages as well). The second, and a sort of logical follow-up to the first, is that the linguistic exchange between Germanic and Latin was almost always from Latin to Germanic, and rarely from Germanic to Latin. Germanic also provides an interesting study of what effect the Romans believed learning a second language other than Greek (or third, I suppose, if they already spoke Greek) could have on a Latin speaker. There is a letter written by a man named Sidonius, writing to his friend Syagrius. Syagrius was a well-educated and well-off fellow, living in a Germanic-speaking area for a time. He remarks on Syagrius’ rapid acquisition of Germanic, and says that he should be careful lest he forget his Latin. He also notes that Syagrius is now an arbiter of disputes among the Germanic speakers, and that Syagrius is almost beneath himself in learning Germanic to fluency. In general, Sidonius reacts with derision towards Syagrius’ learning of Germanic, something which indicates that few Romans learned Germanic, and that they felt (inaccurately) that learning one language would detract from one’s knowledge of another.

To address Gaulish specifically, since it’s included in your original question: there was certainly Latin-Gaulish bilingualism, especially in Cisapline Gaul. Like with Germanic speakers, however, the majority of Gaulish-Latin bilinguals would speak Gaulish natively, with a relatively small portion learning Gaulish as a second language. During the Gallic wars, Caesar would on occasion write letters about military affairs in Greek—implying that if they were captured by the Gauls they would be impossible or more difficult to read than Latin-language letters. (Recall from earlier that one of the areas in which Latin was prominent over Greek was in military matters, so this is an atypical use of Greek). Latin had a greater effect on Gaulish than vice-versa, but loan words came from Gaulish into Latin, especially relating to horsemanship and transport, due to trading connections between native Gaulish and Latin speakers, and relatively large merchant populations in Cisalpine Gaul.

In terms of the actual method of learning the language, anyone learning Germanic or Gaulish would not have done so in a classroom environment, but rather by immersion or exposure. The speakers of languages other than Latin or Greek who had spoken Latin natively almost always resided in an area where that language was spoken, and they learned it through exposure, and not through classroom lessons.