How do scientists figure out all these distances in our universe?

There is a whole branch of astronomy which deals with measuring the distances of objects in space, and it takes whole books to describe the techniques used. The distances to nearby stars is measured using parallax: the amount the star moves during the six months it takes the Earth to move from one side of its orbit to the other. A second technique for medium distances (globular clusters and nearby galaxies) is to use Cepheid variable stars, whose luminosity is directly related to their period of variation. The farthest distances are measured by the red shift (shift of spectral lines towards the red end of the spectrum) caused by the Doppler effect, in turn caused by the expansion of the universe.