Way of Life
A civilization's way of life is dependent mainly on the area surrounding them. The face of the moon would also affect their way of life. Does the surface have many continents? Or is it mainly archipelago? How does the continents looks like? Vast plains? Or mountainous? What is the surfaces temperature? Does it have many tundra? Or desert? See, many factor would affect a civilization's way of life.
Based on our Earth, mountainous region would have many farms due to its fertile lands. Archipelago would depend on the sea as their main source of food. Vast plains means it is susceptible to typhoon and tornado, like America. Since the moon has more land than water, then it would have more desert since the rainfall should be lower than Earth.
Time and Cosmology
The gas giant is the most visible thing they can see on the sky. If they are tidal locking (the moon and the gas giant) then your people would be aware that the land they are on are rotating.
One of the key aspect that caused people in the olden times believe that the Earth was the center and that the sun and other sky objects revolves around the Earth was because they all seems smaller than the Earth. Due to the size of the gas giant, your people would be aware that they are revolves around it. They would believe that the center of the universe is the gas giant, that all things revolves around it.
Thus a year would be the time needed for the moon to finish revolving around the gas giant. Day would be the time where there is sunlight, night would be where there is no sunlight. Also, depending on the composition of the gas in the gas giant, it might be very bright at night due to the gas giant reflecting the sun's light.
Since the moon orbit a gas giant and thus would sometimes have no access to direct sunlight due to the gas giant blocking it, they might develop belief on dark deity that caused the world to be engulfed in darkness for some days/months/years, depending on its revolution angle relative to the gas giant. Since the gas giant is also the biggest thing on the sky, they would also worship it.
As science progress, they would develop more awareness of their home world. Some of the scientist might get persecuted for their discovery.
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