I've looked up here and on Griffths' didatic Introduction to Electrodynamics, but, still, am struggling to understand why time has the opposite sign of the space coordinates. Any insights are welcome.
From the 2nd postulate of special relativity, and homogeneity of space-time and isotropy of space, follow that the space-time interval, i.e. $s^2 = c^2Δt^2 - Δr^2$, between two physical events is invariant. If you change the sign of $Δt^2$ (or $Δr^2$) in the interval, then it cannot be invariant anymore.
One starts special relativity with the idea that the speed of light is the same in every reference frame. From this one concludes using the Pythagorean theorem that time must pass slower for a moving observer. The distance traveled in space $x$ plus the time, that actually passed for you, $s$, is the time $t$ that would be observed from a different observer.
This proper time $s$ then becomes the new measure of things. So you start with $x^2 + (cs)^2 = (ct)^2$ and then define a new scalar product (using the metric tensor) that gives you $(cs)^2 = (ct)^2 - x^2$.
I feel that this is not justification enough. If there were no minus sign, the addition of velocities would not work such that the maximum velocity is the speed of light $c$.
If the time coordinate had the same sign as the space coordinates, then time would be completely equivalent to space. Clearly it isn't - for example, causal influences (i.e. pretty much everything) can only move forward in time, but it certainly isn't the case that everything can only move in one direction in space!