How did the Roman government regard Greece and Greek culture? Good or bad? I have a couple of specific questions because I realize that this might be a too broad topic.
What happened with the Greek religion? Did the Roman religion become dominant or did they blend. In that case how?
What was the situation of the Greek language in ancient Rome? More specifically, did Greek spread within Rome or did Latin overtake Greek in former hellenic colonies. An area which I am particularly interested in is in the eastern Mediterranean. Did Greek spread here during the Roman conquests in Anatolia?
How was Greek litterature regarded by the Romans? Especially among intellectuals.
Did the Rome overtake Greece's interests in philosophy and science?
You would probably be very interested in reading Pliny the Younger's letter to a man called Maximus. This is what I immediately thought of when I read your question.
In this letter, Pliny reminds Maximus, who has apparently been sent to Greece on some official duty, to treat the Greeks with utmost respect, and then lists the reasons why. He tells him to appreciate the greatness of the Greek civilization and keep in mind that Roman legal code and Roman ideas of freedom and citizenship are derived from old Greek customs.
Unfortunately I don't think anything else about Maximus is known. That would give a fuller picture of how the Romans (or at least Pliny) viewed the Greeks in this context. Pliny certainly holds a reverence to the ancient Greek society, but some of his letter reads as if he is instructing Maximus that the best way to work with the Greeks is by respecting them, that they are necessarily deserving of all the respect he has spoken of. Still, I think this letter is very interesting, and should begin to answer your question.