I am curious that if there were a gas giant sitting close to a red dwarf with an Earth-like satellite orbiting it:
How could we determine the way the planet appears in the sky and the way it interacts with the sun as a celestial object?
Or in other words, how does the planet in the sky appear and when would it do so given certain parameters?
I know this is a pretty complex question with a lot of things implied, but I just don't even know where to start for this stuff.
"It depends", as always.
In most plausible situations, you'll find that any moon of a gas giant will be tidally locked to its parent, eg. on a bit of the moon facing the planet, the planet will be in pretty much the same location in the sky at all times. I say "pretty much" because there are effects like libration which I can't fully quantify, but for all intents and purposes if you're on the planet-facing side, it'll always be in the same place, and if you're on the non-planet facing side you'll never see the planet ever. Note that this has some other important knock-on effects, like the fact that the day-length of the moon will be the same as its orbital period, which will almost certainly be at least several Earth-days long, and maybe Earth-weeks. That means it'll have really long nights, which will cause interesting climatic and evolutionary effects.
Given plausible moon orbit radii, the gas giant is likely to appear to be much larger in the sky than the parent star is. For the tidally-locked moon, this means that on the planet-facing side, every "day" you'll see the sun pass behind the gas giant. This will look like an eclipse if the planet is high enough in the sky from your point of view. It will not look like an eclipse of our sun by our Moon, because the gas giant will appear to be much bigger, and will obscure the star's corona.
If by "Earth-like" you meant "Earth-sized, with its own Moon-sized moon" then it is entirely possible that you moon will not be tidally locked (see this answer to a recent question about gas giant moons with their own moons). It is pretty likely that the moon-and-submoon system will be rotating in the orbital plane of its parent though (because the submoon is likely too big to be captured into such a neat orbit) so for any given point on the moon's surface, the parent gas giant will rise and set at pretty much the same time every day and describe the same arc across the sky. During each "month" (the period of rotation of the moon about the gas giant) you'll have an eclipse period, where the gas giant will be between you and the star. Whether or not you'll see an eclipse depends on the length of the moon's day (eg. the period of rotation about its own axis) relative to the length of the "month".
Note that this is a fairly unlikely scenario though.
As to the precise whens and wheres of planetrise and set, and what it would look like and when the star would be obscured by the gas giant, that depends on a huge raft of parameters including but not limited to:
If you wanted guidance on what these should be, you could perhaps ask a fresh question decribing what you want, and asking if it is possible and whether anyone could supply some likely parameters (or if not, what might be the closest realistic setup to your requirements).
There are some planetary system simulators and renderers out there, such as Celestia. Instructing you on there use is beyond the scope of this answer and the remit of this site, but there are guides and examples out there. Wrestling with open-source tools is often a frustrating exercise, but you can't beat actually rendering images of what the view will look like.
