|MadSci Network: Astronomy|
Just from looking at the sky on a clear night, you can deduce that we seem to be in the middle of something big and flat: the Milky Way appears as a fuzzy band of stars and dust extending around the whole sky. This wouldn't happen in a more sphere-shaped galaxy. You can also deduce that most of the stuff in this flat thing is somewhere around Sagittarius, and very little of it is around, say, Auriga. So, just from your backyard, you can also guess that we're not right in the center of this flat thing - if we were, it would look the same in all directions.
You're correct in saying that the details are obscured by dust - but that applies mostly to visual light. Different wavelengths of light - radio, microwave, infared - are able to penetrate the dust quite well; a microwave image of the Milky Way, in fact, shows a beautiful edge-on view of a galactic disk with a bulge in the center.
That's probably as much as we can say without beginning to crunch numbers. To find out details about the structure of the Galaxy, you really do need to map out the positions of individual stars. This is quite difficult; you have to observe stars all over the sky, estimate their distance, estimate how many other stars you are unable to see at that distance ... but the result is indeed a map of the galaxy. It's like a radar-map; you stand in one place and make distance-observations in all directions, and figure out where things are.
The dust indeed is the biggest problem; it was not until 1930 that anyone figured out how to deal with it. You have to estimate how much dust there is; this can be done in various ways, which merit a textbook of their own. You also have to make some assumptions about how different types of stars are distributed. Here's an example: if we look out in some far and dust-obscured direction, perhaps we will only see (and barely see) a single red giant with an absolute magnitude of +1 (i.e., very bright). By looking around our own neighborhood, we know that for every +1, very bright star, there are 50 or so dimmer stars. We can presume, then, that surrounding this faraway red giant, there will be 50 stars that we simply cannot detect. We have to put all 51 of these stars on our galactic map! On a map like this you can see the spiral arms and other features.
Recently, astronomers started making this sort of map of the whole Universe: plotting the positions and distances of as many galaxies as we can see. Surprisingly, although the universe seems fairly uniform as seen from the ground, when viewed in 3D it is structured like a sponge; galaxies are distributed in sheets and bubbles and hairs. This is one of the big mysteries of modern cosmology.
Check out Chapter 14 of Zelik & Gregory's textbook to learn more about our galaxy and its "discovery."
Try the links in the MadSci Library for more information on Astronomy.