Thanks to IR telescopes we can see the center of the galaxy. And through redshift we can measure velocities of other stars and gauge our own velocities. Then a few simple formulas can answer a lot of questions.
But historically, our position in the galaxy caused great debate, with astronomers like Hubble and Kaptain (?sp) making their best guesses. Dust obscured their view of the galaxy's center, and hence caused inaccuracies in their guesses. Google their names for a better explanation.