How early did people understand the size and tangibility of the Moon beyond being a disk in the sky? Were any people aware it was a physical place? Or was it considered more mystical like stars used to be?

The classical “Greek” philosophers were absolutely obsessed with geometry, both theoretical and practical (although the practical came later). Pythagoras and his followers argued for a spherical Earth based on ideal form, which Plato naturally found compelling. Aristotle took things a step further by providing empirical evidence for a spherical (or spheroid) Earth. This is important for several reasons, but mostly because empirical observations about the size and shape of the Earth were used to estimate the size and shape of the Moon.

So, by around 300 BC, philosophers were using geometry and observations (shape of shadow of Earth on the moon during eclipses) to calculate estimates of the relative sizes of the Earth and Moon. This then allowed Aristarchus to estimate the distance from Earth to the Moon (as well as the distance from Earth to the Sun). Eratosthenes (~250 BC) was able to further such calculations, and by the time of Ptolemy (~140 AD) the classical world was very comfortable with concepts such as a heliocentric solar system, the moon as a satellite of Earth, and elliptical orbits.^*

Why are orbits and shapes of satellites important? Because it is clear that by the time of Aristarchus (and probably before), the Greeks conceived of the Moon as a physical, spherical object with an elliptical orbit. The ancient calculations were based on angles, distance between cities, and the calculation of relative radii of Earth, Moon, and Sun. All of these calculations depend on that being measured (the Earth) and that being compared (the Moon and Sun) being bodies.

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The best sources are the surviving writings of the Greek philosophers themselves, and the most accessible are Plato’s Phaedo and Ptolemy’s Almagest. Most of the writings from this period are simultaneously primary and secondary sources, in a way not dissimilar to modern scientific treatises (summarize previous work; post a hypothesis; state methods; do calculations and summarize evidence to support hypothesis; conclusions).

The better sort of history of science and introduction to astronomy courses at universities cover this period and related concepts. One of the better synopses is available from online lecture notes from a UC Berkeley course. I don’t know who the author is, but it is very readable and accurate to the extent of my knowledge.

^* Edit: Ptolemy was actually going against the general grain of Classical astronomy by suggesting a geocentric Universe and circular orbits. Aristarchus and Eratosthenes, among others, had deduced elliptical orbits and Heliocentric models of the Universe/Solar System. Thank you, /u/thejukeboxhero, for helping me to clarify that. I included Ptolemy because by that time it is very clear that Classical astronomers considered "heavenly bodies" to be physical objects in orbit about something. He got the something quite wrong, but his work was pretty amazing, even if operating on what turned out to be poor assumptions!