Only Ancient Greek music has surviving musical notation, the majority from the Hellenistic Period. A lot of these are papyrii but the most famous ones are the Delphic Hymns and the Seikilos Epitaph. You are correct in that Ancient Greek (and Roman) society highly praised music as it was used from everything to funerals, marriages, theatre to epic poetry. We know that the Ancient Greeks had a complex musical system that evolved through time although I won't go into detail because I know next to nothing about musical theory. On the Romans, nothing remains of their musical notation so we can only make educated guesses and reconstructions. It is thought that the Romans borrowed the Greek method of notation and tuned their instruments to Greek modes. They also borrowed or had instruments that were similar to the Greeks such as the tibia/aulos, the cithara (the origin of the word "guitar"), the lyre, a hydraulic pipe organ, a medium or long necked lute called a pandura, small cymbals called Cymbala and a Tympanum that was either a frame drum or tambourine. The Romans were also fond of brass wind instruments such as the Cornu and the Lituus. Both of these were used for signalling in the military but also for processionals. Both the Romans and Greeks thought music was a sign of knowledge and music competitions were very common. If you would like to hear what the music may or may not have sounded like definitely search for Christdoulos Halaris' album "Music of Ancient Greece" and the musical group Synaulia's albums "Music of Ancient Rome" Vols 1 & 2.