Einstein's thought experiment with the clock made of two parallel mirrors is simple as can be. The stuff about ordinary clocks and trains is fairly easy to understand too. That's the way I would go, a qualitative understanding.
The reason SR is so counterintuitive is that it's hyperbolic. Hyperbolic relations just don't occur very often in everyday life. The only example I know is locating a sound using two different listening points. One's brain does this automatically. (Very useful for hunting, that's why it evolved.) So I would teach that.
I once had a dramatic lucky experience with this. My girlfriend inadvertently made a stereo cassette recording in her apartment. The recorder had each microphone directly above its speaker. We played the recording back without moving the equipment. It was amazing realistic. We could hear her walking around the apartment talking and "see" where she was at all times.
A cousin of mine said he heard a similar experiment with the sound of an airplane taking off played back in an anechoic chamber with the speakers in the right places. The airplane appeared to pass by directly overhead.
People try to get that effect with headphones but I'm told it doesn't work particularly well. People have different bodies so in real life they get different reflections of the sound.