The expansion into Gaul, Britain, and Germany and the expansion into the eastern Mediterranean are different in a variety of ways, so I’m going to address the latter and hope someone with more expertise in the former can come along and complete the discussion.

Put simply, the expansion east was generally framed by the Romans as “peacekeeping” and assisting allies. The Hellenistic east was a hotbed of dynastic expansion and competition in the centuries after Alexander’s expansion. Roman involvement in Greece, Asia, Syria, etc., in this period usually begins as an ally giving aid, sometimes but not always with Roman involvement continuing after the threat has been dealt with. Eventually, political involvement morphs into full-on administrative control, but it rarely seems to have started that way. Interestingly, treaties with Rome from this period tend to be initiated by the other party, but are also quite standardized, suggesting that Rome had a high level of influence in their drafting.

Maintaining peace in eastern regions was of interest to Rome as a way to keep expansionist rulers from setting their sights on holdings closer to Italy (as Macedonia did in the third century), and because instability in coastal regions was resulting in piracy, which impacted trade. Early moves east, starting in the third and second centuries BCE, included defending Greece from Macedonia, and preventing the expansion of Armenia into central Turkey. Eventually in the first century, Pompey Magnus, who had distinguished himself in Roman civil wars and by eliminating piracy from the Mediterranean, was given administrative authority over all of Rome’s dealings in the east. Rome already had administrative control over some places, but Pompey’s reputation, skill, and physical presence in the region allowed the Senate to solidify their hold over Asia and Syria.

Israel is a pretty informative example of the ways that Roman influence could grow from “friendship” to political influence to direct or indirect administrative rule. In the second century BCE, after winning the region from Seleucid control, Judas Maccabeus made a treaty of friendship with Rome, recorded or referenced in 1 Maccabees 8. There doesn’t seem to have been any material benefit following this treaty--Rome didn’t send troops to defend against Seleucid armies--but it certainly would have sent a message. Rome didn’t care a lot about the region in this period, but they did care about limiting Seleucid power and, not much later, in maintaining stability after the fall of the Seleucid dynasty. In the first century BCE, the Hasmonean dynasty in Israel had fragmented, and Pompey was invited down from Syria to adjudicate the dispute. It’s a long story (Wikipedia’s summary in the Pompey article is good), but it didn’t go as either of the Hasmonean factions would have liked, and Pompey ended up reducing Hasmonean control to the region of Judea, and putting the northern parts (Galilee, Samaria, Idumea, etc.) under the control of Syria--which was already client to Rome. A few decades later, Herod the Great would use his connections with the Roman imperium to take control of the whole area again as a client king. Notice that at every point, Roman influence at least nominally responded to invitation by the local rulers, and military action could arguably be framed as either directly provoked or necessary to prevent greater warfare. That is, according to surviving sources, most of which are pro-Roman.

Documents from these periods are incredibly eloquent on the benefits of Roman control over the barbarous easterners. Cicero’s speech in favor of the Manilian Law, which is the law that gave Pompey all that authority I just mentioned, waxes for pages about Roman losses to Mithridates and the necessity, for the safety and economic security of Rome, of having a firm handle over eastern affairs. The way that the word “freedom” is used by many pro-Roman authors with reference to eastern expansion is nothing short of propagandistic--as long as a city wasn’t controlled by the other guys, Rome considered it “liberated.” Never mind that most of these “free” cities now paid tribute to Rome. Modern historians still use words like “pacify” to describe what Rome was doing, and we still paint resistors as “bandits” and “pirates.” Unfortunately we don’t have those peoples’ perspectives to help defend them.

A couple of good histories of Rome’s eastern expansion are Mediterranean Anarchy, Interstate War, and the Rise of Rome by Arthur Eckstein and Hegemony to Empire by Robert Kallet-Marx. They both emphasize this centuries-long creep of influence, with the military as a promise and a threat, as opposed to viewing it as “conquest” in the strictly military sense.