So, after thinking about the subject for many years, I've written a book on Relativity! It is called "Relativity Made Real", reflecting my hope that it can make the phenomena of Relativity seem more concrete and real to people, rather than difficult and abstract.
Here is the link: http://www.relativitymadereal.com.
Essentially, what I do is approach the topic in a very physical way, explaining how the phenomena arise from the underlying, mechanical properties of matter. I do not start from mysterious "postulates" nor from abstract "spacetime" concepts (although these are discussed in the second half of the book).
Rather, I start from the fundamental nature of electromagnetism and other "field theories", and show how this leads to the effects of Relativity. In this way one gets a very concrete picture of why moving clocks run slow, why moving things get shorter, and why energy and mass are interchangeable. Then one is in position to understand the meaning of Einstein's "postulates", and also the origin and significance of the spacetime concept.