What is the correct order of transformations scale, rotate and translate and why? This is a rather primitive question coming from an electronic engineer. When applying rotate (about origin), scale (in which we shall translate towards origin and then back) and translate, does it matter in what order we do it? Why?

Basically in my case I have an image in coordinate space that goes from -2 to +2 in x and -1.5 to +1.5 in y to get ration of 4:3. This range comes from how the original mandelbrot set is plotted. I need to scale and translate this to fit into an axes (of pixels) in bitmap that goes from 0 to 800 in x axis and 0 to 600 in y axis. I am trying to understand what matrix to use to scale and translate the points from my mandelbrot set into the bitmap image.
Usually you scale first, then rotate and finally translate. The reason is because usually you want the scaling to happen along the axis of the object and rotation about the center of the object.

In your case you don't really need to worry about this generic solution though, but you only need to map range [0, 800] $\rightarrow$ [-2, 2] for x-coordinate and [0, 600] $\rightarrow$ [-1.5, 1.5] for y-coordinate, in order to map screen coordinates to real/imaginary components for Mandelbrot calculation. So this is simply done by:
$$real=4*(x+0.5)/800-2$$
$$imag=3*(y+0.5)/600-1.5$$

Note that you need to calculate Mandelbrot coordinates from screen coordinates and not the other way around. This is to ensure that you evaluate the Manderbrot equation for each pixel exactly once and are not left with holes in the image or do double evaluation per pixel.