Well, Rome came to control the Greek peninsula in 146 BCE after defeating the Corinthians and their allies in the Achaean League (which was a confederation of the Greek city-states) at the Battle of Corinth which was the definitive battle of the Achaean War fought between the Roman Republic and the Achaean league, although certain city-states retained partial independence.

The Achaean War came at the end of Rome's effective annihilation of its rival Carthage (146 BCE) and the conquest of Macedonia in the Fourth Macedonian War (defeated in 148 but became a province in 146), following these accomplishments Rome began to act with increasing hostility towards the Achaean League and demanded that Sparta, which had always been a reluctant member of the League and not very well-liked among the city-states be released from the confederacy, but the Achaean league refused and chose a war they knew they had little hope of winning instead of bowing to pressure from Rome and Sparta. Now, the Achaean League and Rome were not always hostile, they were actually allies due to cultural and religious affinities between the two and aided each other in several wars, but this did not last as the Achaeans considered siding with the Macedonians in the Third Macedonian War 171-168 BCE and tensions began to grow. Someone might flesh out the details of the war a bit more because that is not going to be my primary focus here.

As for how Greece impacted Rome in the years following conqest, it is hard to overstate the profound impact it had on Roman literature, culture, and religion. As Horace famously said "Graecia capta ferum victorem cepit", Greece took its wild conqueror captive. The Greek language was for much of antiquity the lingua Franca of the Mediterranean, particularly in the east, but it became the favoured language of the scholarly, the aristocracy and the elite in Rome, and the writings of poets such as Homer or philosophers such as Plato and Socrates became popular.

Greek mythological and legendary tradition became an even more important part of Roman literary traditions, and many famous individuals such as Cicero, Scipio Africanus, and a good deal of emperors were philhellenes who had a fascination with all things Greek. One corollary you could equate it with would be Anglophilia or Francophilia in their colonies, as it was considered the height of sophistication and taste which came from a centre of civilization. Many Greek traditions and religious practices had already been a part of Roman tradition from Etruscan influences and the impact of the Syballine Books.

Art and statues are visibly impacted by Greek artistic stylization and design, scenes from Greek stories and histories become more popular and nudity which was not often depicted in Roman statues became more common.

Roman theatre was heavily impacted by Greek drama, and Fabula Palliata is a genre of Roman theatre based on Greek drama, usually taking the form of comedies but are not so by definition, other subjects were included in the genre such as tragedy, some more famous writers in the genre include Terence and Ennius.

One of the best sources on this period is Polybius, who I want to talk about because he has such an interesting story, he was one of 1,000 Achaean nobles who were taken as hostage by Rome following the Third Macedonian War as punishment for the League involvement with Perseus of Macedonia, he became a member of the house of Lucius Aemillius Paullus Macedonicus the man who conquered Macedonia and educated his sons, and was a counsel of Scipio Africanus during the Third Punic war and returned to Greece following the defeat of Corinth to use his Roman connections to ease the post-war conditions in Greece.

I am sorry I do not have more time to write more extensively but I am busy with exams, hopefully I will be able to add more in an hour or two but maybe someone else can.