Here’s a simple explanation of how the Moon moves:

• It orbits the Earth: The Moon is held in orbit around the Earth by gravity. It takes about 27.3 days to complete one full orbit relative to distant stars (sidereal month), and about 29.5 days to go from one New Moon to the next (synodic month), due to the Earth’s own motion around the Sun.
• It also rotates on its axis: The Moon spins on its axis once roughly every 27.3 days, which happens to be the same time it takes to orbit the Earth. So we always see roughly the same face from Earth (near side) — this is called tidal locking.
• Why the phases change: As the Moon orbits, different parts of it are lit by the Sun. When the Sun-lit portion we see grows, it’s a crescent; when more of the Moon is lit, it becomes a full Moon; and when less is lit, it becomes a waning Moon.
• orbital plane and eclipses: The Moon’s orbit is inclined a few degrees to the Earth’s orbital plane around the Sun. Eclipses (solar and lunar) happen when the Sun, Moon, and Earth align near the orbital nodes.
• What moves how: The Moon’s motion is a combination of its orbit around Earth and the Earth’s orbit around the Sun. The daily rise and set of the Moon is mostly due to the Earth’s rotation, while the Moon’s gradual drift against the background stars is due to its orbit.

If you’d like, I can share a Jain perspective on celestial movement as described in traditional Jain cosmology as well.