In order to measure up the expanding universe, astronomers need to know the distances to cosmological objects over a large range of values. Overviews of how this is done and what is found are given on this page and its download.
The following distance measuring techniques that are used are discussed - in engineering style, of course:
Measuring how apparent position of a nearby object shifts relative to very distant objects during the Earth's travel around the Sun.
ii) Proper motion
Two methods of determining distance by means of proper motion: the moving cluster method and the statistical parallax method are discussed.
iii) The "standard candles"
"Standard candles" means sources with known absolute luminosity. Two types are considered: Cepheid variable stars and supernovae type 1A.
The three above methods are in principle all that is needed to get a good "distance ladder", but there are many more methods, all trying to improve accuracy at certain distance bands.
The attached pdf file will give you good idea of how astronomers measure up the expanding universe by means of the distance ladder.