Did the Romans take medicine from Greece?

In a way, yes. The Greeks developed the theory of the four humours, which the Romans continued, and introduced the follow on theory of opposites. Both empires were observers and spent alot of time observing the sick, and it is easy to think that the Romans only copied the Greeks, because they were the first to discover many theories.

The Romans had many more new ideas than the Greeks in terms of medicine; Sanitation, Sewage system, etc.

Other than that, they shared very similar views. They both had herbal remedies, doctors, amputations, theories, basic surgery, cleanliness, and both believed strongly in the gods.