The types of vegetables and their relative amounts are variable, and can be based on whatever vegetables you have and want to use. What matters is that you have more or less 5 cups of mixed vegetables to begin with.
Starting with onions and garlic is always a good place, and certain types of vegetables, like carrots, parsnips, root vegetables in general, and hard squash, are good additions, giving body to a blended soup without need of added flour.
As to cheese, I used a hearty smoked provolone In this version, but feel free to use your own taste or available ingredients to guide you...with a preference for medium soft cheeses that melt well.