We don't know very much about the initial redpills who started each Zion or what they thought. We can piece together some elements of those early days but not much. The best we know is from Morpheus:
When the Matrix was first built, there was a man born inside who had the ability to change whatever he wanted, to remake the Matrix as he saw fit. It was he who freed the first of us, taught us the truth: ‘As long as the Matrix exists the human race will never be free.’ After he died the Oracle prophesized his return and that his coming would hail the destruction of the Matrix, end the war, bring freedom to our people.
The Matrix (transcript)
Morpheus doesn't explain how the first were unplugged, so we don't know what the first redpills would have believed about how they were unplugged. If they believed that the man who unplugged them (the previous One) "had the ability to change whatever he wanted" then presumably they just assumed he could unplug them using his own abilities (i.e. without help from the machines, which might hint to them the role of the machines in the cycles of the One).
Morpheus' statement does indicate that the first redpills were taught that the One would return, so they evidently had some notion of their role in searching for the One. The redpills were taught by both the previous One and the Oracle; the Oracle's role in instructing the first redpills is indicated by both the previous quote and another of Morpheus' explanations to Neo:
Neo: So is this the same oracle that made the, uh, prophecy?
Morpheus: Yes. She's very old. She's been with us since the beginning.
Neo: The beginning?
Morpheus: Of the Resistance.
The Matrix (transcript)
The only other information about the first days of Zion that I'm aware of comes from some hints from The Miller's Tale. In it, Morpheus is a boy who is told a story about one of the "earliest" redpills, a man named Geoffrey:
Geoffrey was responsible for cultivating the first wheat for Zion as a more palatable food than the porridge they normally had to eat. The comic hints at the extreme hardship faced by the earliest members of Zion but doesn't say much about what they thought other than the fact
that they were trying to destroy the Matrix:
Of course, we already knew that the first redpills were taught they they must destroy the Matrix from Morpheus in The Matrix.
It's also possible that the first redpills attempted to reconstruct the true past by exploring the real world. In Hunters and Collectors (H/T @Valorum), we are introduced to a collector redpill named Flint who was an archaeologist in the Matrix. Flint explored the ruins of human cities on the surface of Earth because, he says,
we weren't finding enough around Zion to distinguish what in the Matrix was created by people and what was invented by the machines.
Flint found a variety of artifacts, including books. Clearly the redpills were interested in trying to learn about the past, so it's possible that in some iteration(s) of the Matrix they found something which was left behind by the redpills of a previous iteration. We don't know for sure if the initial redpills were able to do any of this exploring, though -- the early days of each Zion iteration may have always been too dangerous for the initial redpills to do any exploring.