The walk along the inner ramparts (lices) — between the two rings of walls — is included in the Château Comtal ticket and is the best way to understand how the fortifications work. The outer wall is lower and older (much of it late Roman, in alternating bands of stone and brick), the inner is taller and thirteenth-century, and the corridor between them is a defensive killing zone.
You can also do a free outer circuit at ground level along the broad grassy lices, all the way around the city. An evening walk here, with the walls lit and the Aude valley spread out below, is one of the great things to do in Carcassonne.