I am your nightmare poster - a layman trying to learn special relativity. (I'm also a refugee from stackoverflow.com: trust me, it's only going to get worse). Apologies in advance if my question is nonsense or too open-ended. I won't be offended if you vote to close it.
I'm working through Taylor & Wheeler 'Spacetime Physics'. The book starts with a nice analogy of different teams of surveyors mapping out a world with different sized measuring rods (and even the same teams using different sized rods for different coordinate axes), and coming to realize that there is an underlying structure which is independent of the coordinate system. The idea is that spacetime intervals likewise give us the objective measuring stick by which we can map out spacetime (presumably by some process analogous to triangulation).
I'm just not able to intuit spacetime intervals like this, especially that distances in time and space have opposite sign. I can draw the diagrams and answer the exercises by plugging numbers into essentially the same equations over and over. Nor do I have any trouble working with the different inertial observers with all their meter rules and clocks. It's just the God's eye spacetime view which is eluding me. If intervals are real and objective, it ought to be possible to project this lattice of intervals onto the world and see it independently of any frame of reference (just like I can imagine the earth without the surveyors' grid lines superimposed on it). I accept that it might be some funny shaped space like a tennis ball turned inside out, but still...
So, my questions are: do you ever get a proper intuitive model for Minkowski spacetime, so that you can visualize the intervals, move objects about relativisticly in your mind's eye, and so on? If so, are there any tricks or insights which helped you get there? Do things 'really' move in your model, or is time just another static dimension like in the Minkowski diagrams?