León is compact enough to walk everywhere that matters. The old quarter — cathedral, San Isidoro, Barrio Húmedo, Plaza Mayor — takes about 15 minutes to cross on foot. San Marcos is a 15-minute walk west from the old quarter across the Río Bernesga.
MUSAC and the newer parts of the city are about a 20-minute walk from the cathedral, or a short taxi ride. There's a municipal bus network but it's rarely necessary for the sights.
Cycling is practical on the flat central streets. The Camino de Santiago trail through the city is well-signed and useful as a walking route even if you're not a pilgrim — it threads through the most historically interesting parts of the old quarter.