By taking the northern coastal route the drive will mostly be highway, the A-8 (some toll portions). Once you turn south onto the N-621 the road will be two lanes and twist and turn through the mountains.
The inland route would take longer, but you could make stops in either Lugo, Ponferrada, Astorga, or Leön for some sightseeing. Again, most of the way would be highway, A-6, AP-71, A-60, with the section by León (AP-71) being toll. The short toll portion can be bypassed by taking the N-120. Just east of León you’d go north on the two lane N-625 to N-621 to go to Potes. Again, twists and turns going through the mountains.
The speed limit on the N roads will mostly be 70-80kph (42-56mph); slower in the mountains. There are plenty of speed cameras in Spain so ensure you stay at or below the limit.
Cangas de Onis, as mentioned by ekscrunchy, is a pretty town and very near the national park. From it, it’s easy access to the park.