The centre of Blois is small enough to cover on foot — the château, cathedral, old town, and river are all within a 15-minute walk of each other, with the proviso that the upper town involves stairs and steep streets. Comfortable shoes matter more here than they do in most French cities of this size.
For the surrounding châteaux, the easiest options are a rental car (several agencies at the train station), the seasonal Rémi shuttle bus to Chambord and Cheverny from the station in summer, or the Châteaux à Vélo cycle network — over 400 km of signed cycle routes connecting the great châteaux, with bike rental from shops in Blois and most other Loire towns. Chambord is about 18 km by bike along quiet roads and forest tracks.
City buses (Azalys) cover the outskirts but are rarely useful in the centre. Taxis are easy to find at the station.