Roman expansion began with its leadership of a confederation of allied cities, the Latin League. Rome led this coalition against nearby hill tribes that had periodically raided the area, then against the Etruscans that had once ruled Rome itself. Just as the Romans started to consider further territorial expansion, a fierce raiding band of Celts swooped in and sacked Rome in 389 BCE, a setback that took several decades to recover from. In the aftermath, the Romans swore to never let the city fall victim to an attack again.
A key moment in the early period of Roman expansion was in 338 BCE when Rome defeated its erstwhile allies in the Latin League. Rome did not punish the cities after it defeated them, however. Instead, it offered them citizenship in its Republic (albeit without voting rights) in return for pledges of loyalty and troops during wartime, a very important precedent because it meant that with every victory, Rome could potentially expand its military might. Soon, the elites of the Latin cities realized the benefits of playing along with the Romans: they were dealt into the wealth distributed after military victories and could play an active role in politics so long as they remained loyal, whereas resisters were eventually ground down and defeated with only their pride to show for it.
Rome rapidly expanded to encompass all of Italy except the southernmost regions. Those regions, populated largely by Greeks who had founded colonies there centuries before, invited a Greek warrior-king named Pyrrhus to aid them against the Romans around 280 BCE (Pyrrhus was a Hellenistic king who had already wrested control of a good-sized swath of Greece from the Antigonid dynasty back in Greece). Pyrrhus won two major battles against the Romans, but in the process he lost two-thirds of his troops. After his victories, he made a comment that “one more such victory will undo me” - this led to the phrase "pyrrhic victory," which means a temporary victory that ultimately spells defeat, or winning the battle but losing the war. He took his remaining troops and returned to Greece. After he fled, the south was unable to mount much of a resistance, and all of Italy was under Roman control by 263 BCE.
It is important to emphasize the extreme militarism and terrible brutality of Rome during the republican period, very much including this early phase in which it began to acquire its empire. Wars were annual: with very few exceptions over the centuries the Roman legions would march forth to conquer new territory every single year. The Romans swiftly acquired a reputation for absolute ruthlessness and even wanton cruelty, raping and/or slaughtering the civilian inhabitants of conquered cities, enslaving thousands, and in some cases utterly wiping out whole populations (the neighboring city of Veii was obliterated in roughly 393 BCE, for example, right at the start of the conquest period). The Greek historian Polybius calmly noted at the time in his sweeping history of the Republic that insofar as there was a deliberate intention behind all of this cruelty, it was easy to identify: inspiring terror.
Roman soldiers were inspired by straightforward greed as well as the tremendous cultural importance placed on winning military glory - nothing was as important to a male Roman citizen than his reputation as a soldier. Likewise, Roman aristocrats all acquired their political power through military glory until late in the Republic, and even then military glory was all but required for a man to achieve any kind of political importance for the vast majority of elites.
The overall picture of Roman culture is of a society that was in its own way as fanatical and obsessed with war as was Sparta during the height of its barracks society. Unlike Sparta, however, Rome was able to mobilize gigantic armies, partly because slaves came to perform most of the work on farms and workshops over time, freeing up free Roman men to participate in the annual invasions of neighboring territories. One prominent contemporary historian of Rome, W.V. Harris, wisely warned against the “power worship” all too many people have succumbed to over the centuries when studying Roman history - Rome did indeed accomplish remarkable things, but it did so through appalling levels of cruelty and astonishing levels of violence.