Sorry, I do realise that this is probably a really stupid question, but can someone please help me understand the following statement:

My understanding of a timelike path, is one which always stays inside a light cone, defined by lines with gradients c and -c that cross at the origin.

What does it mean to say that time coordinate itself is timelike or spacelike (and the same for space-coordinates?) And what does it mean to say "x plays the role of time".

To find the nature of any coordinate, let that coordinate vary while holding all the other coordinates constant (fixed). This results in a curve in spacetime. At any event p on the curve, the tangent vector to the curve at p gives the nature (spacelike, lightlike, or timelike) of the coordinate at p. Note that even if both p and q are on the same coordinate curve, the nature of the coordinate does not have to be the same at p and q.

You can also define a timelike path in terms of the metric--use the metric to do a line integral of ds^2 along the path, if it's spacelike the result should be positive, if it's timelike the result should be negative. And as George Jones said, a small increment of the time coordinate (while holding other coordinates constant) results in a path through spacetime as well. Likewise a small increment in the x coordinate will result in a different path, and if this path is timelike, that's what it means to say "x plays the role of time".

Unfortunately there is no universally agreed convention which way round this should be; different authors do it different ways. If you're not sure of the author's convention, remember there ought to be 3 spacelike coordinates (all with the same sign for ds^{2}) and 1 timelike coordinate (with the opposite sign) (unless there are null coordinates for which ds=0).