we would like to explore independently some the different sections of the Great Wall.

I wonder if your destination is Beijing. If so, these Wall sections are generally located in the suburb area and the public bus may not be a convenient choice to shuttle between them. You are recommended to charter a car. But some of the sections are connected and you can hike from one to the other. For example, you can walk all the way from Jiankou section to Mutianyu.