Hi,
I'm a guide used to do this multi-day trip.
On this route there are plenty of interesting places to visit, you have two options, to follow the coastal path, to Santiago, where you will pass through the beautiful Santander, where nearby you have exciting places, like Santillana del Mar, the Altamira caves, Comillas with one of the few buildings created by Gaudí. The fabulous Picos de Europa, all the coastal towns of Asturias, (Lastres, Cudillero, Gijón) Cathedrals Beach as you enter Galicia, then you can work your way to Santiago stopping in Lugo famous for its complete Roman walls, and then Santiago.
Option 2: Following the "Camino de Santiago", with many stops, like Logroño and the many places along the way like Burgos, Leon, Astorga,( both of these last ones with buildings by Gaudí), Ponferrada and the world heritage site of Las Medulas, once in Galicia, the Ribeira Sacra, and then Santiago and beyond to Finisterre.
I do recommend renting a car or getting a private driver-guide.
I hope this helps you.
Juanjo