Sorry if this is a silly question. I know that one of the major pieces of evidence for the universe expanding is red shift – the doppler effect causing light waves to appear redder due to relative motion. Is it possible for us to measure differences in red shift to get a (vague) notion of where we are in relation to the rest of the universe?

Never apologize for an earnest question!

There are two parts to a redshift: the doppler effect which is due to the relative motion of us and the emitting object, and the Hubble expansion of the Universe, which stretches the space between galaxies and galaxy clusters.

So we can determine the overall expansion of the Universe, and ask, of all the leftover stuff once we subtract that out, how fast we’re moving with respect to everything else. (We can’t determine our deviation from an ideal position; not enough information exists for that.)

The answer is that the Solar System moves relative to the leftover glow from the Big Bang at 368 ± 2 km/s, and so the local group —  the Sun, the Milky Way, Andromeda and all the others  —  are moving at 627 ± 22 km/s relative to the CMB.

Long article about our cosmic motion here.