The most common way of testing is to look at how the contents of the universe show signs of the geometry of the universe. Because, through general relativity, gravity works through geometry, looking at the overall geometry of the universe and the change in that geometry over time give us information about the contents of the universe and vice versa.
So, for example, looking at the way that light behaves as it reaches us from far away gives us information about the expansion of space.
Other ways of testing tend to be more indirect methods of the first. For example, looking at the distribution of light elements in the universe (hydrogen, helium, and lithium) gives us a restriction on the possible history of the universe, including the past geometry and the possible changes of that geometry in the past (since the changes control how much hydrogen would have turned into helium and lithium).
Looking at the way structures form in the universe also looks at the overall geometry. The way that clumps of matter break away from the whole to form galaxies and galaxy clusters depends on what the overall amount of matter is and how that influences the overall geometry of the universe.
I like to recommend Ned Wright's cosmology tutorial on matters like this. http://www.astro.ucla.edu/~wright/cosmolog.htm