This is not my area of expertise, but one of my areas of expertise is related, and since there are no other answers here, I can at least tell you what I do know.

As always in music, practice leads the way and theory explains it later.

The Greeks did not "discover" the "various scales of Western music." Music is physics, and the arrangements of Western scales do not exist in nature as we play them today. (the physically perfect fifth, perfect octave, and the overtone series DO exist in nature, and the Greeks did know this.) What we have now (scales, keys, etc) began to develop in the early baroque era when modal composition began to go out of style and the world of music--composers, theorists, and even instruments-- struggled to accommodate the newest developments. This was a really interesting time because Platonic education and academies were all the rage, which means that the biggest players in this society strove to connect themselves to the ancients in every way they could. (Why? that's an interesting question with a very interesting set of answers, but not your question, so I'll not go into that.)

So what we call the scales of Western music have their roots in early baroque music, which was looking to the ancient world for both inspiration and justification. Throughout western music history, theorists created different interpretations of ancient musical treatises to try to justify and explain what they were doing in their own present day. But very generally speaking, in the medieval era, there were modes based on ideas of ancient music (not actual ancient music) which functioned not as scales or keys but as whole-step/half-step relationships within hexachords with melodic motives unique to each; in the Renaissance these modes developed further to adapt to the growing demands of polyphony; and in the early Baroque era the concept of keys was just beginning to be useful, which later brought scales as we know them today. In ALL of these eras, theorists read ancient treatises and tried to use them to explain what they were doing.

One thing that DID happen in 17th-century Rome (so not the Roman music you asked about, but--) was that theorists like Doni went so far down the rabbit hole of ancient music that they began building replicas of ancient instruments like lyres and monochords. I can tell you more about ancient theory and how it was interpreted in the 17th century if you want, but it is very technical and I'm not sure if you want to read about it. (If you do, let me know! I'm happy to tell you about it.)

So: our current ideas of "modes" (Dorian, Phrygian, etc.) are misinterpreted medieval modes, which are misinterpreted ancient modes. And our current scales are far from anything in the ancient world.

There is some (very little) surviving ancient music, like the Seikilos epitaph, but I can't tell you very much about it, because it is just not my area. What I did learn in my studying is that ancient music that is performed today is heavily interpreted and full of guesses, because the short answer to your question is: we don't know enough to know anything for sure.

One composer who might interest you is singer-songwriter Sappho.

In Everyday:

Music was a part of Roman culture from earliest times. It was customary at funerals, and the tibia (Greek aulos), a woodwind instrument, was played at sacrifices to ward off ill influences. Songs (Carmen) were an integral part of almost every social occasion. The Secular Ode of Horace, for instance, was commissioned by Augustus, and performed by a mixed children's choir at the Secular Games in 17 BC. Under the influence of Greek theory, music was thought to reflect the orderliness of the cosmos, and was associated particularly with mathematics and knowledge.

Influences:

Etruscan and Greek at first, and during the Imperial Period they took a little bit from everywhere. Africa, Asia Minor, etc. We can assume they borrowed notation from the Greeks and also measured scales in the same method of Greek modes.

Instruments:

They used a great deal for many different purposes. I won't list every one.

There was the tubus (Latin "tube") which was a military horn used for trumpet calls on the field.

The cornu (Latin "horn") was a long tubular metal wind instrument that curved around the musician's body, shaped like an uppercase G. It had a conical bore (like a French horn) and a conical mouthpiece. The cornu was used for military signals and on parade.

The tibia (Greek aulos - αὐλός), had two double-reeded (as in a modern oboe) pipes, not joined but generally played with a mouth-band or capistrum (Greek phorbeia - φορβεία) to hold both pipes steadily between the player's lips.

The lyre, borrowed from the Greeks, had a sounding body of wood or a tortoise shell covered with skin, and arms of animal horn or wood, with strings stretched from a cross bar to the sounding body.

They also had percussion, ranging from large timpani to small metal cymbala, or symbols.

In Summary:

Everything Romans knew about music they essentially took from Greeks. Instrument design, notation, and theory, to name the biggest parts. There were composers that adopted Greek theory to write pieces for Roman festivals, and Romans made use of music in all aspects, from military trumpet calls to parades and games.

Sources:

Scott, J. E. 1957. 'Roman Music' in The New Oxford History of Music, vol.1: 'Ancient and Oriental Music,' Oxford: Oxford University Press.

Habinek, Thomas. 2005. The World of Roman Song. Baltimore: Johns Hopkins University Press.

Bonanni, Filippo. 1964. Antique Musical Instruments and their Players: 152 Plates from Bonanni's 18th-Century "Gabinetto armonico"