If I recall college correctly, I was taught that the Romans essentially idolized the Greeks, as shown by the Romans copying the Greek's religion and a lot of their artwork. More recently, I was told the Romans actually looked down upon the Greeks. They saw them as manipulators, liars and a lazy people who were not fit for combat. This claim was backed up by a source that looks pretty dubious to me, so I figured I'd ask here. What were the Roman's attitude toward the Greeks?
Did the Roman's attitude change towards Greece as Rome grew? For instance, did the Romans of the Republic period think of the Greeks differently than the Romans of the Empire period?
Also, I know the Western half of the Empire (ie, the Byzantine Empire) was predominantly Greek, and I'd imagine they may have had a significantly different view of themselves than the Eastern Empire may have. Is this the case?
copying the Greek's religion and a lot of their artwork
The statement is not exactly on point, since the Romans didn't so much consciously copied Greeks so much as being heavily influenced by the Etruscans to their North(who were in turn heavily influenced by the Greeks). Rome developed under Greek influence, which led to them sharing many similarities in their culture and having some kind of mutual respect, not just culturally, but also in certain military traditions. Pre-maniple Roman army employed phalanx similar to the Greeks, an aspect which remained during the Manipular formation with the presence of Triarii. The Romans adored the Greek glorious battles during the Persian wars, which gave way to the popularity of Corinthian helmets in Italy, eventually causing there to be a Italian variant(look up italo-corinthian). And as a result of such adoration and cultural sophistication, Romans didn't look down on Greek the same way they branded their Gauls and Germans barbarians.
During early Republican period, Rome allied itself with the Arcadian league in an attempt to exerts its influence over Greece and wait for eventual conquest. Around 2nd Century BCE, I believe they were successful in controlling nearly all of Greece. They didn't receive kindly to Greek uprising, which led to the destruction of Corinth, but they treated Sparta kindly, using it as a tourist attraction because of their renown.
Romans and Greeks as such have mutual appreciation, but like any other groups, even when they have things to like from each other, still would consider themselves superior, which is where your idea that Romans regarded Greeks as weak might have come from, although still, I don't see that very often. And from the way Polybius received Romans, it's hard to imagine there are too much cultural hostility rather than political. It is kind of important to note that the Greeks employed a very different tactic on foreign affairs than Rome. While the majority of Greek city states were mostly interested in their own development and usually exerts their control over colonies indirectly and often for the development of their home city, the Romans are a much more ambitious breed who had no qualms about annexing states and controlling lands. Virgil's propaganda work declared that Romans have a destiny to rule the world, but curiously also implied that Greeks are natural enemies of Rome, since Romans are descended from Trojans
From Tactitus's brief account of Germanicus in Greece, it's evident that the recently(200 years) Romanized Greeks considered themselves part of Rome, but retained some kind of Greek identity. By the time Eastern Empire become a thing, Rome has become a Imperial state that focused less on individual ethinic identity and more on the collective recognization that they are Romans. (Byzantine is to the East, fyi). Byzantines see themselves as Romans, although I am not very qualitifed to talk about the extent to which they retained their Greek identity.