To get the right answers, you must ask the right questions. In this case, the question is not, "How does Relativity work? What is it?" The right question is "How does motion work? What is it?"
Motion is caused by a standing wave in spacetime, because it must be. Einstein showed that if two things are indistinguishable, then they are identical. (Which is also generally considered to be common sense.)
If relativistic effects are caused by both motion and by gravity, then these two effects must have the same cause. That cause is the stretching of spacetime. It really is this simple in concept.
I can't draw, so I'll "borrow" an internet picture about the Alcubierre drive (which can't work for incredibly obvious reasons now that we've proven the existence of gravity waves moving at the speed of light).
Stolen shamelessly from trekmovie.com
This is how motion and special relativity work. The particle is somewhere in the center region. Spacetime is warped around it as a standing wave. Relativistic effects depend on the warping, which depend on the angle you're viewing it from. Don't forget that you have your own warped spacetime, so you have to adjust for that. The math is no more difficult than high school trigonometry, because the theory of relativity is nothing more than the very careful application of the Pythagorean theorem of right triangles. (A squared plus B squared equals C squared.) Things going forward (right) gain energy from the gradient (blueshift). Things going backwards (left) lose energy to the gradient (redshift). Things going exactly sideways don't change at all.
Please remember that this is a 2.5 dimensional picture of a 3 dimensional effect. The faster the object moves, the more the field will be tilted/warped around it.
If two objects are moving at the same speed in the same direction (at relative rest), they will not see any relativistic effect, because the losses for one are exactly cancelled by the gains in the other along the straight line between them. Any change in speed or direction by either body will show a relativistic effect, because then the two bodies are no longer at rest relative to each other, and the math no longer cancels out to zero.
Note that spacetime is a field of potential energy governing motion. The gradient is literally distance over time. Note that space is flat/Euclidian (a^2 + b^2 = c^2), while time is hyperbolic (a^2 - b^2 = c^2). This complicates the math in general relativity, but you can more or less ignore it when dealing solely with special relativity.
Special side note - A particle's spin rate is internally constant, no matter its speed, no matter the local strength of gravity. This means that spin self adjusts to exactly match relativistic effects. This strongly suggests that spin is, in effect, an expression of time.