What is a simple argument to prove that the stars in the sky are further away from the Earth than the Moon?
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
Lunar occultations. Just missed the moon block Mars last month. Not sure when Mars will block a star, thereby proving the stars are further than the moon.
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.