Stars more or less move all at once. Someone looking at the sky over time could notice some stars seemingly moving in separate directions to the rest of stars that also happened to appear brighter. Eventually they realized these things were separate bodies orbiting the sun.
This Wikipedia page is pretty cool, as it not only goes over the history of when they were discovered, but even mentions what the astronomers saw that tipped them off in the first place: [https://en.wikipedia.org/wiki/Timeline_of_discovery_of_Solar_System_planets_and_their_moons](https://en.wikipedia.org/wiki/Timeline_of_discovery_of_Solar_System_planets_and_their_moons)
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