Eclipses: Since you're not tidally locked, your moon is going to be a fair distance from the gas giant. This means that while it will still be very large and a dominant presence in the sky, it probably won't be too insanely large. Because it's a fair ways away, you don't need to worry much about the orbit - even a relatively small inclination will prevent most eclipses (the gas giant will pass under or above the sun instead of through it). You could design the orbit so you get regular eclipses, but that's unlikely to occur naturally. When eclipses do happen, they will likely be longer than on Earth (I suspect the gas giant will be larger in the sky than Earth's Moon), and you will be able to see the sun through portions of the gas giant's atmosphere, but the majority of it will fully block light.
Tides: The gas giant will be the dominant presence on the tides, but your sun will still have some impact. Tides will probably be similar to that of Earth, but with larger differences between low and high - having more of the tides from one source should increase the amplitude of tidal frequencies.
Geologic activity: You will probably get more volcano and earthquake activity than on Earth. It won't be by an incredible amount, like if you were tidally locked, but the presence of a close gravitationally dominant body is going to stir things up more than what happens on Earth. This will mean some geologic features, like mountain peaks, might be more exaggerated compared to what we see on Earth. There will still be a lot of variation, though - an older mountain range will still be worn down, it's just that the newer peaks might be higher.
Weather: I am not qualified to answer this definitively, but I suspect that you will get some weather generated by the gas giant's tidal forces. So you might get more extreme storms. This will interact with other climate forces though, so I have no idea how it will turn out - it might be more hurricanes and stronger storms. Or it's even possible that the gas giant could be at the right frequency to act as a moderating influence, where it actually reduces the power and frequency of storms.
Navigation: Your people will be able to use the gas giant as a navigation aid. Importantly, they'll be able to do this for portions of the day, as well as at night, and the gas giant is hugely visible compared to stars. The combination of the gas giant + either stars or the sun will make navigation easier, although it won't be trivial - ships will still need things like charts and time keeping, and overland navigation will still require some level of expertise.
Natural Life Cycles: on Earth lots of natural cycles tie roughly to lunar orbits. I suspect on the moon of a gas giant, any native life might have cycles that tie to the orbital period of the moon. Depending on the distance, this could be a much greater or smaller period (although not too much smaller since you're not tidally locked).