RIP_s:
A key concept you seem to be missing here is that the redshift we measure is very, very small. We talk about "cosmological redshifts" as if they're Huge. They are, in fact, so tiny they're extremely difficult to measure.
The consequence of this is that the visible wavelengths of light appear to your eye to be unchanged. The light from a galaxy is broadband (it fills the electromagnetic spectrum). The way we measure its redshift is to disperse that light into a spectrum and look at the spectral lines of elements whose positions we know (from analyzing the light from those elements in a laboratory). This calibrates our spectrograph so that when we find the shifted lines we can calculate how much they've shifted.
When light comes from a massive object, it also is gravitationally red-shifted (the light has to "climb out of the object against its gravity", if you will). So a correction factor must be applied for that, as well.
The cosmological redshift occurs because spacetime itself is expanding. One way to think about that is to imagine that the "ruler" we use to measure large distances in space is stretching, as well. That allows the different light components to be measured as if the galaxy were standing still, relative to one another, except for the gravitational and rotational components of the redshift. Our challenge is to find the amount by which the ruler has stretched: the cosmological redshift.