Method to observe clocks on spaceships moving near $c$? I have read the explanation for clocks slowing down on a spaceship moving relative to a stationary observer -- something like a beam of light between two mirrors taking longer for the observer to bounce back and forth just because of the distance whereas the person aboard the ship does not see this effect because he does not see himself as moving. But my question is, how in practice would the observer actually measure this effect and is a procedure for doing so important to this thought experiment?
 A: I think it is important.  For instance if it turned out that it was not possible to observe this effect, even in principle, then there would be legitimate questions about whether the effect mattered -- akin to the questions that arise about various proposed mechanisms 'underlying' Quantum Mechanics.  So it needs to be possible to observe the various clocks other than 'by magic', and that needs a mechanism.
Secondly, showing how clocks can be observed lets you understand the considerations when making a measurement, and in particular various things which have to be taken into account.
So, here's one way of doing the measurement: you have the clocks you want to observe emit some kind of pulse of light or other EM radiation every time they 'tick'.  Then various observers simply count the pulses arriving and compare their timing with a train of pulses from their own clock.  Note that this is not dissimilar (although much simplified from) to what happens in real life: a GPS satellite, say, is emitting some EM signal which encodes its clock's 'ticks' among other things and a GPS receiver is listening to it (the GPS receiver generally does not have a good local clock, of course, and GPS is much more complicated than this).
And this way of doing the measurement tells you something important: because the clocks are moving relative to each other there will almost always be Doppler effects, and these effects need to be removed to expose the underlying relativistic effect: when you hear pitch shifts as cars pass you this is not because the clocks that are driving their horns are running slow from your perspective, it's because of the Doppler effect.
So knowing how the measurement can be made, if in a simplified way, both makes it clear that it can be made, and makes clear what artefacts such an experiment will need to remove to see the underlying effect. 
A: The concept of an "event" in special relativity is, well, special.  An event is a particular point in space and time, i.e. spacetime, where something happens. The key idea behind "relativity" is that event can have different space and time coordinates (labels) for different observers, and that those are physically meaningful.
So you're asking an important question. How do you measure what two moving-relative-to-each-other clocks say at two different points?  
In one frame (the one you think is stationary), you'll pick points (t=0, x=0) and (t=1second, x=v meters):  The origin and where the other clock will be one second later.
In the other (moving to you), you'll arrange for the first point to also be labeled (0,0). You want to measure what the clock says at the second other one.
There are lots of ways to do it. Taylor&Wheeler, a old book that many people learned from years ago, suggested having the second hand of the moving clock sticking out perpendicular to the motion and having it punch through a piece of paper, so by seeing where the hand punched you can read the time it said as it went by.  A more modern approach is just a flash picture of the clock as it goes by. The key is to measure it right there as the clock goes by.
Note that SR is really about the entire coordinate systems:  Things that are in one place and/or time in one frame aren't in another.  You should learn to think about those separate spacetime points separately. Thinking in terms of "What I see over here from things over there" tends to be more complicated, and can lead to lots of subtle paradoxes about simultaneity. The classic pole-in-barn paradox is one example.
A: Sometimes picture that shows moving light clock is not complete, probably that's why you didn't like it. I believe that "good" picture must include two Einstein - synchronized clocks of a "stationary" observer
Yes, an inertial clock moves very fast and never comes back to the observer. The observer needs two identical clock at some distance apart from each other.

We can demonstrate time dilation of the SR in the following experiment (Fig. 1). Moving with velocity  $v$ clocks measure time $t'$. The clock passes past point $x_{1}$ at moment of time $t_{1}$ and passing past point  $x_{2}$ at moment of time $t_{2}$.
At these moments, the positions of the hands of the moving clock and the corresponding fixed clock next to it are compared.
Two Einstein - synchronized clock measure greater time interval, than the "moving" one.
More practical way as the user @tfb very correctly noted is to measure Transverse Doppler Effect, or simply measure frequency of the moving source of radiation (light emitted at points of closest approach). You will measure that radiation will be $\gamma$ times reddish, since all processes in the moving lamp (source of radioation) run slower.
In laboratories researches put source of radiation on the rim of a cenrifuge and spin it very fast Absorber in the center of the centrifuge (or vice versa) measures the Transverse Doppler Effect or deviation of frequency of the source.
