I have been reading Hawking's 'A Brief History of Time' and it has gotten me thinking about Einstein's theory of relativity, in that it assumes that an object must have infinite mass if it is to be traveling at the speed of light (please correct if I'm wrong in my beginner's knowledge of physics).
But, do light waves have any sort of measurable mass? Or in that same vein, do sound waves? Is it somehow possible?
The particles of light waves - the photons - have the rest mass $m_0$ equal to zero. However, at the speed of light, $v=c$, the total mass $$ m= \frac{m_0}{\sqrt{1-v^2/c^2}} $$ is increased to an indeterminate form, $0/0$, which should be evaluated as a finite number. The photons - and everything else - carry the total mass that is proportional to the total energy via the famous $E=mc^2$ relation.
Yes, this mass may be measured. For example, uranium nuclear power plants burn the uranium and reduce its mass by 0.1 percent or so because the waste products (the nuclei) are actually a little bit lighter. This energy may be completely transformed to the radiation coming from light bulbs - and the light from these light bulbs carry 0.1 percent of the uranium mass away. This mass is a source of gravitational field and adds inertia to boxes with this light etc.
Sound is different. The speed of sound is much smaller than the speed of light.
While "phonons" in low-temperature condensed matter physics - particles of sound - are analogous to photons in many respects, and $E=mc^2$ still applies, the same is not true for sound waves in the air etc. Because the temperature of the air is nonzero, the "ground state" - the lowest-energy state at fixed conditions, with the minimum number of "sound quanta" or "phonons" - is not really unique. Instead, there are many states of the air "without any sound" which correspond to chaotic configurations of the air molecules. So one can't consistently divide the energy of the air to the energy of its ground state and the energy of the phonons.
But of course, if you produce some loud sounds, they will carry lots of energy in the air and the mass of the air will inevitably increase by $m=E/c^2$ which is, well, not too high because $c^2$ is a large number.
an object must have infinite mass if it is to be traveling at the speed of light
No, that's not true at all. An object's mass is a fixed property that doesn't change, regardless of what speed it travels at. But its energy does change. The energy increases with increasing speed, according to the formula
$$E = \frac{mc^2}{\sqrt{1 - v^2/c^2}}$$
Here $m$ is the mass, an inherent property of the object, and $E$ is the energy. What you're probably thinking about is the fact that an object that has a nonzero mass ($m > 0$) can never travel at the speed of light, because it would require an infinite amount of energy for it to do so. But zero-mass particles like the photon (the quantum of light) can travel at the speed of light without requiring infinite energy. (In fact, they can't travel at any slower speed.)
Some people use the term "mass" to mean the quantity that I'm calling energy (in different units), and the term "rest mass" when they want to refer to what I call the mass. In that case, one would say that a material object traveling at the speed of light would have infinite mass. I refer you to another answer I've written for more information on the historical context of the terms.
But, do light waves have any sort of measurable mass? Or in that same vein, do sound waves?
No, but they do have energy. For light waves, the energy is related to the frequency $\omega$ of the wave,
$$E = n\hbar\omega$$
($n$ is the number of photons), and for sound waves, the energy is related to the amplitude (particle displacement) $\xi$ and the frequency $\omega$,
$$E = A\rho \xi^2\omega^2$$
where $A$ is the cross-sectional area of the sound wave.
You could calculate an "equivalent mass" as $m_\text{eq} = E/c^2$, which would tell you the amount of mass it would take to have the same energy (at rest) as a given light or sound wave. If it were possible to convert the energy of the wave into mass directly, $m_\text{eq}$ would be the amount of mass you'd get. But that's the only sense in which a wave has mass.
Sound waves affect atoms , use them as a pathway, but the atoms in any of the three standard forms do not give any mass to the energy that affects them , i.e. Sound waves. The road you drive your car on lends no mass to the car your driving
I believe this question is not worded correctly only because the submitter is not that well versed in scientific terminology.
To put it briefly - neither sound waves nor light waves have mass, that is because they are representations of moving particles, they themselves are not entities and do not exist beyond the scope of human-facilitating terminology. As an analogy in a similar way one could ask "Does traffic have a mass"?
However photons - the particles which create and constitute light waves have mass, albeit a curious one, on the other hand the particles that carry/produce sound waves have quite a usual mass, since that are all the atoms of the material world.
Remember - photons make a light wave, but a sound wave can be made/carried by any atom.
