[spiderfarmer]

Correct me if I'm wrong, but I asked my astronomer brother in law some time ago:

We can't look at the Milky Way but we can look at similar galaxies. Our current understanding of what our own galaxy looks like is based on measuring the shape and direction of (groups of) stars. Stars contain(ed) Hydrogen and Hydrogen emits a radiation with a specific wavelength, that we can use to detect it. With Doppler Shift we can measure direction, if the emitting atoms are moving away, the wavelength is slightly larger. And they'll be slightly less if they're moving towards us.

Stars orbit the center of our galaxy and if we can get their orbit radius we can calculate how fast they are going. With enough measurements in all directions and using a lot of trigonometry, we can get to a map.

[acqq]

As far as I understand, the Doppler shift can't be used to measure the distances of the stars in the Milky Way, but it is used to estimate the distance of the farthest galaxies, billions of light years away from the Milky Way. My other post here has the links about the parallax, the method that measures the actual distances of the stars around us. And then there's the Cosmic distance ladder:

https://en.wikipedia.org/wiki/Cosmic_distance_ladder

which includes the methods you described.