I'm aware that it is best to use a variety of greens to make compost for my lawn. How about browns? Can I use only leaves or only newspaper/cardboard or only sawdust, etc, or is a mixture more effective?
Newspaper and cardboard will contribute very little in nutrients, so where they are used other things such as leaves should be added. The reason for a variety of materials is to get a wide range of nutrients since each plant tends to accumulate different nutrients in different amounts. The more diverse the mix the more nutrients your compost will have.
The best composts have hundreds of foodstuffs in them.