The impact that Greece had on Rome can be best summed up in the famous words of the Roman poet Horace who said that "Captive Greece took captive her rude conqueror." In other words, practically everything about Roman society and culture showed Greek influence.
Greek culture impacted Roman culture as Rome conquered Greece. Roman military men sent back art and literature that they looted from Greece. Many Greek slaves were sent to Rome as well. Because sine if them were well educated and cultured, they tended to be used as tutors to wealthy Romans.
In these ways, Greece was able to have a major impact on Rome. Greek ideas influenced Roman architecture and Roman literature. They influenced Roman philosophy and Roman education. They even were able to play a huge part in influencing Roman religion and mythology, with Roman gods and Greek gods coming to be associated with one another.
Basically, it is impossible to think about Roman culture without thinking about the ways that it was influenced by Greek culture.