The Romans had an interesting relationship with the Greeks: although Romans had long captured Greece and places where Greek culture thrived (like Tarentum in Southern Italy or Corinth), the Romans found that their culture was still widely under the influence of the Greeks. There were many Greek dialects in Rome, and as Greek writer Polybius famously said, Rome was under the inevitable exposure of Greek culture- the Magna Graecia.
The main reason for this exposure is due to the relative lateness of Roman culture- existing Greek literature began around the time of Homer and dates back to the 8th c. BCE. Classical literature soon began in the 4th c. BCE, with the period characterised by tragedians, orators, historians and philosophers like Socrates or Plato. Then began the age of Hellenistic or Alexandrian literature.
The birth of Latin literature was around 240 BCE, when Rome emerged victorious from the Punic Wars. One of the earliest writers was Livius Andronius, who adapted Greek plays for the Roman theatre and translated the Odyssesy in Latin. This happened write after Corinth was sacked and Quintius Flaminus declared "freedom and self-determination of all Greeks".
Centuries later, Horace would write about this in his famous Epistle- "Greek, once conquered, conquered its savage conquerer and brought its art to rustic Latium" (or, in Latin "Graecia capta ferum victorem capit et artes intulit agresti Latio."). Cato the Elder described the Greeks as "vile and indocile" and said how Greek literature would corrupt everything in famous letters to his son. However, he himself used lots of Greek literature to improve himself, so this could suggest his hypocrisy and ambiguity of opinion on the Greek problem.
Cicero, however, took on a different approach as he mentioned the Roman indebtedness to Greeks, or at least to the Greeks as they once were, not the "Graeculi"- the conquered Greeklings. He lamented in his In Defence of Flaccus "Greece-now weakened and virtually broken is supported by the reputataion of that city." He also wrote about the poverty of words in the Latin language and how some words simply cannot be expressed in Latin.
However, a few decades later, Rome would find itself in the Golden Age of Latin literature, where people like Caesar, Catallus, Lucretius, Horace, and many more would show the world the beauty of the lingua Latina.