Downtown Durango is compact and walkable — Main Avenue between 5th and 13th streets is the whole core. The Trolley is a free city bus that runs the length of Main throughout the day; the same agency runs a paid local bus network reaching Fort Lewis College and the surrounding neighbourhoods.
A car is useful but not essential for staying in town. For day trips — Mesa Verde, Silverton via the road, the Animas wilderness — you'll want a car or a guided tour.