How do we know, without using modern equipment, that the stars are further away than the moon in the night sky? Further, is there a simple and actionable argument to prove that this is indeed the case? Additionally, I would like to know how people in earlier times knew this fact without having access to modern equipment, including telescopes.
If you don't mind modern equipment, there are some arguments. For example, we have sent spacecraft out well past the moon to get to other planets. As you approach a planet, the direction to it changes. Eventually, there it is beside you instead of in front. But the stars have mostly not changed their apparent direction at all.
Every year, the Earth goes around in an enormous circle. The size is something like the distance to the nearer planets. You would expect to see nearer stars move past the more distant stars. Something like the nearby trees appear to move past distant mountains when you drive on the freeway. And this has been seen. But the motion is tiny. It takes extremely careful measurements to see it. The biggest shift is less than 1 arc second = $1/3600$ degree. Given the size of Earth's orbit, this puts the nearest star at about 4 light years away, much much farther than the moon and planets. Most stars are so far away that no change can be seen.
Modern research has shown that stars are bright like the sun. Given that they appear to be faint little dots, they must be far away. First because something so bright appears to be so faint. And second, something so big looks like a point, not a disk.
Edit - Measurements with early telescopes in the 1600s
As Charles Francis' answer says, the Hipparchus calculated the distance to the moon is $60$ $1/2$ Earth radii, and the distance to the Sun is $2550$ Earth radii. More measurements became possible after the invention of the telescope. For example, Galileo discovered 4 moons of Jupiter in 1609 or 1610.
Wikipedia article on Romer's determination of the speed of light says that in 1671 - 1676, Romer made the first crude measurements of the speed of light, and used it to find some distances. Predicted times of eclipses of the moons of Jupiter were off by up to 15 minutes. He attributed this to the time it takes light to travel from Jupiter to Earth. The time varies as the orbits of Earth and Jupiter bring them closer together or farther apart.
He found light takes much less that 1 sec to travel an Earth diameter. The time to travel from the Sun to the Earth was 10 - 11 minutes. This confirms the Sun-Earth distance is much bigger than the distance to the moon.
The distance from the Sun to Jupiter was part of the calculation. Romer only had crude Earth-Sun distances, so his Earth-Jupiter distances were also crude. This website gives the Earth-Jupiter distances used by Romer as 3.95 to 5.75 Earth-Sun distances. Again, much bigger than Earth-moon distances.
The fact of parallax in the observed position of the moon was known in ancient times. This makes possible a calculation of the distance of the moon in terms of Earth radii. No parallax was then observed for the Sun. The first known calculation of the distance to the moon is generally attributed to Hipparchus of Nicaea
We have used radar ranging to determine the distance to the Moon, the Sun, Mercury, Venus, Mars, Jupiter, the Galilean satellites, Saturn, the rings of Saturn, and Titan. We have fired radio signals at stars and have not observed a return signal. Either stars are closer than the moon and somehow invisible to radar (which would be very surprising for a ball of plasma having the blackbody radiation spectra we observe from stars), or stars are much further away.
Passing back in time a bit... Hipparchus (ca. 140 BCE) used the shadow of the Moon on the Earth during a solar eclipse to estimate that the Sun is about 2500 Earth radii (getting the ratio of Moon distance to Earth radius within about 10% and getting the ratio of Sun distance to Moon distance 10-times too small).
Ptolemy (ca. 150 AD) used an epicyclic model of planetary motion to lower bound the distance to the celestial spheres to more than 20,000 Earth radii (which is bazillions of times too small).
So the stars are at least 8-times further away than the Moon.
Compare angular sizes of stars and sun. Sun has angular size of $\approx 32 \,'$ and our closest star Alpha Centauri A has angular diameter of $0.007\,''$. This gives of angular size ratio between them as about $270\,000$ times, which correlates very well with real distance ratio between them and Earth. You don't need a measurement device for this. You can see sun's visual diameter by putting some dark black glass in front of eyes. And stars are merely like "dots" when we look at them at night. Because sun is just an ordinary G-type main-sequence star, it's not that hard to conclude that other stars in a night sky is far far more distant than our sun.
Now moon has comparable angular diameter with that of sun. It would be unwise to expect that moon mass would be greater or comparable to that of sun, thus it's natural to conclude that moon can only be at a distance closer to Earth than that of our sun. So by concluding it we imply that stars are also a lot more distant than moon.
If you take it as a given that the Earth is revolving around its axis once per day than you could make the argument that anything that does not move (after subtracting the apparent movement due to the Earth's rotation) must be far away, or it would fall down.
Satellites have a short orbital period, the Moon a longer one, everything else must be "very very far away". Of course things are complicated by Earth's orbit around the Sun; if that's not a given then one is stuck with Ptolemy's curly trajectories for the planets, but the general idea still holds.