I have a spacetime in 2-dimensions. x is the position and t is the time.
1) t is in nanoseconds and x in feet, so the straight lines may represent 2 opposite waves that overlap and move with the light's velocity. Is that correct ?
2) What's the meaning to say : we may determine the transformations of this 2d spacetime which preserve straight lines and preserve the speed of light ?
Thanks so much
Compare your setting to the one of Newtonian mechanics, where one works in $\mathbb{R}^4$ with one time and three space coordinates. How can you change your inertial frame of reference? Well, any change should preserve time intervals, spatial distance of simultaneous events and motions of free particles. It turns out that you can translate or rotate your frame of reference, and you can move it by constant velocity. This group of transformations is also known as Galilean transformations.
Now, in special relativity you work within the $2$-dimensional Minkowski space, i.e. $\mathbb{R}^2$ with the "inner product" $(t,x) \cdot (t', x') := -t \cdot t' + x \cdot x'$. In general one would of course use $\mathbb{R}^4$, but let's just ignore that.
Now you do the same thing as in Newtonian mechanics: You think of properties that should stay the same when you change to another inertial frame of reference. You definitely want to preserve the light cone $-t^2 + x^2 = 0$, as this is basically the mathematical analogue of "the speed of light is constant". If you want to determine the group of transformations that preserves the light cone, it turns out that Galilean transformations are not suitable for this, but the Lorentz group (or Poincaré group) is!
A useful reference may be chapter 2.7 of "Peter Szekeres – A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry".