Intuitive explanation for why the Doppler effect (and red-shift) happens?
Your intuition is correct - a moving source emitting wavefronts periodically will be closer to the previously emitted wave in the direction of motion, and farther from the previously emitted wave in the opposite direction - see the simulation here.
You are also correct that the size of the effect depends on the speed of the observer relative to the speed of the wave.  That is why we can easily experience the Doppler shift for sound in everyday life (e.g. the sound of a car when it is approaching vs. receding) but we never notice the Doppler shift for light.  The speed of sound is a lot slower, and thus a lot closer to the velocities of everyday life, so the effect is larger for sound.
While the shift for light is too small for our eyes to perceive (e.g. not enough to change an object’s perceived color), sensitive instruments can measure extremely small changes in wavelength.  For example, the spectrographs that astronomers use to make radial velocity measurements of stars for the purposes of detecting extrasolar planets (via the star’s motion as the planet orbits) can now routinely measure velocities with a precision of 1 m/s - a leisurely walking speed for a human!