This is a nice one for visualization:
Going deeper than that, I'd just look for a cheap textbook on relativity that mentions simultaneity in the table of contents, or read Einstein's math-free book on relativity:
http://www.amazon.com/Relativity-The-Special-General-Theory/dp/0517884410
(One reviewer there is saying that that version is missing the crucial images, so make sure you find one that has them.)
One thing I suspect you're getting hung up on is that this ought to make time travel a possibility. It doesn't.
Essentially, events can be separated in two ways: timelike and spacelike. So you should think in those terms instead of simultaneous or not simultaneous.
If events are spacelike separated, then they can be seen as either occurring simultaneously, A before B, or B before A.
If events are timelike separated, then event A precedes event B, always.
The difference between the two types of events is simple. If light has enough time to make it from one event to the other, they are timelike separated.
If light doesn't have enough time to make it between the two events, they are spacelike separated.
This is actually why faster than light drives ALWAYS implies the possibility of time travel, no matter what hand-waving warp drive, hyper drive, or what ever a sci-fi author tries to introduce. If you can travel between two spacelike separated events, you can travel backward in time. They are identical.
Put another way, with spacelike separated events, you can ALWAYS find a reference frame where the events occurred simultaneously, but you can NEVER find a reference frame where the events occur at the same place.
With timelike separated events, you can ALWAYS find a reference frame where the events occured in the same place, but NEVER a reference frame where they occurred at the same time.
I didn't say c, the speed of light in a vacuum relative to an inertial reference frame (which is a frame in uniform rectilinear motion), is not a constant, just that objects can exceed it in a non-inertial reference frame, as can light itself. You point out, correctly, that distant stars constantly exceed it relative to Earth, and this is a perfectly valid system of reference, albeit not the only one.
Special relativity tells us that motion is relative between inertial reference frames, but that the acceleration of the Earth is absolute because it's always being pulled by the gravity of the Sun. General relativity tells us that there are no inertial frames, acceleration is relative too, and gravity is a fictitious force. For a good introduction to relativity, I recommend Einstein's own <em>Relativity: The Special and the General Theory</em>, in which he aims to make the subject as accessible as it can be made to readers without advanced mathematics; I hope you like trains, because Einstein will ask you to imagine a lot of them.
The bottom line is that geocentrism is a valid frame (though, again, not the only one), and any supposed physical proof that the Earth is not at rest rightly belongs in the Einstein was wrong pile. This is old news to physicists, and when I took Physics IV, it was expressly taught in a lecture on relativity that Galileo and the geocentrist establishment "were both right." I say this not because I think it should make a difference to Christians if the Earth moves, or if it doesn't move, or if there is no fact of the matter, but just because it is a scientific error to say that there is any physical distinction between motion of the Earth and relative counter motion of the universe.
When you say "clock in motion", you really mean in motion relative to an observer. The distinction is important, as it is the reason you can observe the time dilation phenomenon.
Example: I have a clock, and you have a clock, which are synchronized to begin with. You and your clock board a spaceship which after a relatively brief period of acceleration (this is to simplify the math), is traveling at 0.5c (that's 0.5 times light speed) relative to me. Now lets say that your spaceship passes by me, and I have a way to compare the ticks of your clock with those of mine. I will observe that your clock is ticking slow. However, you on your spaceship, at rest relative to your clock, will observe that my clock is ticking slow.
So which of us is right? Actually, we both are. By using the Lorentz transformation, we can convert our times to agree with each other. What it really means is that time can only be measured relative to frames of observation. There is no universal flow of time. Time and space are linked.
This is just one part of special relativity, and if you are interested in learning about it, I recommend Einstein's Relativity. It is written for the non-scientist and can be clearly understood, albeit with some difficulty and patience. If you really want to understand it, take a class.
Edit: spelling and grammar
Why not start with Einstein's own book on the subject? Less than $10. Amazon link.