I know that you can calculate the distance to another star for example, by using the phenomenon redshift to determine how much it has redshifted because of the expansion of the universe. But how do you know what the original frequency/wavelength of the light was? (And if all of my above statements are wrong, how do you measure the distance?)
Red shift is used for measuring the distance to very distant stars (galaxies mostly, in fact). The secret is to use spectral lines. Specific elements when very hot emit light at very specific colours and you can spot the pattern of those colours when the whole pattern has been shifted towards the red and see how far it has shifted.
For closer stars, there are other techniques, using types of star whose brightness can be worked out in some way, or, for the closest, by observing how the position of the star compared to more distant ones changes as the Earth moves from one side of the Sun to the other.
There are various ways of measuring distances in the universe which depend on the range in which we are measuring.
Parallax method: Up to 100 - 1000 ly
Exploiting period luminosity relation of Cephied variables: Up to 3 - 10 Mpc
Tully Fisher relationship: Intergalactic distances
Hubble parameter: Large scale structures in the universe
Magnitude measurement of type 1a supernova
Apparent brightness is one way of measuring distance; the further away a certain type of star is, the fainter it will be. We have Cepheid variables whose intrinsic luminosity is indicated by the period of their variability, so we know with certainty how bright they really are. Most stars (as opposed to galaxies) are too close for cosmic redshift to be a factor, so redshift or blueshift indicates their proper motion away or towards us.