There are some really good books on introductory relativity, my dad had one that I read when I was about 12, then again when I was 14 and again when I was in college. You seem really receptive to the concepts, so I'll respond.
You can start with only one fundamental: The speed of light (in a given medium) is constant in all frames of reference. No matter what, the speed of light is always constant.
I have no physics background, or any real knowledge. Here is what I "think" I can take away from this conversation... someone please correct me if I am wrong:
Two objects are moving in opposite directions, each at .75 the speed of light... in reality, the distance between them would be increasing at almost the speed of light, because there speeds would be combined, however due to both objects having mass, their combined speed still cannot reach the speed of light.
Yes, but not exactly for the reason you describe.
If two photons were moving away from each other, each at .75 the speed of light, then they would really be moving away from each other at exactly the speed of light due to them having no mass.
This is hard to say yes or no to. A photon begins its existence at the speed of light and ends it the same way. Time really has no meaning to photon. It boils down to the time dialation of a particle equals it's mass divided by the the square root of the quantity, speed of light minus its velocity. This is zero divided by sqrt(c minus c) or zero over zero. The c's may be squared, but it doesn't matter, it's still zero over zero.
Essentially, everything breaks down *at* the speed of light, so we can only discuss what happens close to the speed of light.
If two objects are moving away from each other, each at .75 the speed of light, then they would appear to be moving away from each other at 1.5x the speed of light from an observer.
No.
Is this even close, or am I missing the whole point?
I'm sorry, writing this I realized I don't have nearly the time, nor the *skill* to explain this in a reasonable post, but please don't think I ducking the issue. There is a *ton* of stuff on the subject on the 'Net. Google "special relativity paradox" and grab a snack because you'll be in for a really enjoyable next three hours.
