To answer the first question - figuring out you're orbiting a gas giant would be hard. While they're still in the Aristotelian model of the cosmos (that is, the cosmos is fixed and rotates around Earth/whatever planet you're on) they simply wouldn't notice. They might observe that the paths of stars are very complex - from the perspective of such a moon, the positions of other planets in the solar system would change wildly over the course of a day - but on Earth we just dealt with that by introducing epicycles (little miniature crystal spheres that allowed particular stars to travel out of sync with the rest of the sky). It wouldn't be until their version of Copernicus or Galileo came along that they'd spot it. But for that, all they would need would be an imaginative mathematician with an eye to the sky and possibly a telescope. By observing other planets in the solar system at different times of the day, this mathematician would notice that the only reasonable explanation is that his vantage point is rotating, with a certain very large radius, which he could then compute; to do this, he'd need to be able to see at least two other planets, so that he could solve for his distance from each and the radius of his own rotation. Now, as rek pointed out in a previous answer, there are several low-tech ways to determine the radius of the moon - by the time our Copernicus came along, the radius would be well-known. He would notice that it is much smaller than his calculated radius - and with a burst of inspiration, would realize that this must mean that the "planet" is orbiting some central point, which itself is orbiting the sun.
Now, our Copernicus might, out of concern for elegance, posit that the point we're orbiting is actually another body; most likely he'd suggest a second Sun. I suspect that with or without this suggestion, the absence of a concrete explanation would make his theory less appealing to the public; heliocentrism would take considerably longer to catch on than on Earth. To prove that there's something there, you need someone like Newton - someone who can come up with a theory of gravity. With a theory of gravity and its effect on the cosmos, you can justify the existence of something this moon orbits, and thereby deduce that there's something there. You would not be able to determine whether it was a gas giant, a star, a black hole, whatever; for that, you'd have to look.
As for the seasons: As rek pointed out, if you don't tilt the system there won't be any seasons. But there will be a killer day/night cycle - at dusk, the moon will be significantly further from the Sun than at noon. I'd think of this like having a miniature winter hit every morning and evening. It wouldn't help, either, that the other side of the moon isn't getting much sunlight, so would be in perpetual winter and would no doubt cool the rest of the moon. To fix this, I have a couple of suggestions: have the gas giant radiate heat (easy enough, if it's large enough to nearly be a brown dwarf star) or just cover the moon with lots and lots of water. Water has extremely high specific heat, so it resists drastic temperature changes - the heat of noon would stick around well into the night. I think the moon would still be a bit colder than we're used to, or intolerably hot in the day if you put it too close to the Sun, but it would be livable.