Lets see if this helps any. Let's consider, for the moment, that you are in the flat space-time of SR. You hopefully already know, that there is an invariant interval, called the Lorentz interval between two nearby events P and P' in space-time, with events being points whose position is specified both in space and time. If P has the coordinates t,x,y,z and P' has the coordinates ##t' = t+\delta t, x' = x+\delta x, y' = y+\delta y, z'=z + \delta z## you can write this invariant interval as ##L^2 = \left( \delta x \right)^2 + \left( \delta y \right)^2 + \left( \delta z \right)^2 - \left( \delta t \right)^2##, which is a quadratic form, and that this interval will be the same between any two observers.
Since you are interested in experiment, let's see how you might find this invariant interval experimentally, without necessarily knowing what the coordinates are - so we'll attempt to talk about what the above means without using coordinates. Basically any experiment you can do is going to take some time to perform the measurement, so you'll have to work around this issue. If you consider the above formula you realize that what you need to do is measure the distance between P and P', square it, find the time difference between P and P', square that, and subtract the squared time from the squared distance. Moeller suggests that you consider a frame in which both P and P' are at rest to perform these measurements. It seems to me that that isn't strictly necessary, though it's convenient if you want to specify a particular way to measure it, in general the rules of SR allow you to use any frame to measure the Lorentz interval - its independent of the observer.
I won't go into the details of how you measure distance, except to say abstractly you can do it with radar, though this process, when you do it experimentally, takes time which you have to account for. Measuring the time difference is similar, it's important to note that you need to use Einstein's midpoint simultaneity convention or some convention that's equivalent to this to determine the notion of simultaneity in whatever frame you choose (Moeller suggest the rest frame) to perform the measurement in. It would get tedious for me to specify in great detail how you go about measuring distances and time differences, hopefully the idea is familiar enough that going into this sort of tedious detail would be unnecessary (as well as boring). The only mildly tricky part about this is to be sure to use Einstein's simultaneity convention to determine time differences.
So hopefully this is sufficient to give you some idea of how you might measure the Lorentz interval between P and P' experimentally in special relativity, without necessarily having a coordinate system and using the coordinate based definition earlier, by measuring the distance between P and P , squaring it, subtracting the square of the time difference, and taking the square root.
What happens when we go to GR? Experimentally, if P and P' are close, nothing is different. The reason nothing is different is that the effects of curvature are second order - it's similar to the way that the Earth's surface is curved, but you can use flat maps for your local neighborhood without any appreciable errors. So as long as you make sure P and P' are close, you just use the same techniques as you did before.
Of course, to be able to talk about measuring the metric at all, you need some operational way of assigning coordinates to events, a procedure that allows you to measure at what coordinates an event occurs at.
It isn't particularly obvious how you'd do this, but for the sake of answering the question, let's just say you have some way of doing it. Given that, to find the metric, you take a bunch of nearby events, and measure the Lorentz interval between them. You then find a curve-fit that expresses the invariant Lorentz interval between two nearby events as a function (which will turn out to be a quadratic form) of the coordinate differences. Why is this function a quadratic form? Well, we already know it was a quadratic form in SR, and because we are considering only small displacements, we expect only linear transformations, and the linear transformation applied to a quadratic form yields another quadratic form.
To spell this out in more detail, if you have generalized coordinates p,q,r,s for some point P, and coordinates p', q', r', s' with ##p' = p + \delta p, q' = q+\delta q, r' = r + \delta r, s' = s + \delta s##, then you have some function of ##\delta p, \delta q, \delta r, \delta s## that gives you the Lorentz interval between P and P'.
The coefficients of , the quadratic form that fits or generates the function f are then just the metric coefficients you were trying to measure.
But it appears that this situation is a bit artificial. We don't actually ever have a very large array of events whereby we measure the distances and time differences between them to "measure the metric". Instead, this is a theoretical framework, and we instead look at things we can measure, like the Shaipiro time delays, light bending, the precession of orbits, that we predict from our theoretical framework, (which includes some prediction of the metric coefficients), and what we do experiemntally is to make sure that our theoeretical framework gives the correct predictions for the simpler things we do measure.
Also note that the values of the metric coefficients are really a reflection of the coordinate choices, the part that we assumed we had some way of doing, without specifying exactly what that way was.
