North Coast of Spain West of Bilbao (to A Coruna)

Charming as FEVE is, a car is really the most practical option to travel anywhere west of Bilbao.

From East to West I have the following in mind:

Cantabria

• I do not like the coast between Bilbao and Santander. The highway runs close to it for most of the way so you'll have good views!
• Santander itself is not the most interesting Spanish city, to me it is skippable.
• Santillana del Mar, Comillas and San Vincente de la Barquera are interesting towns west of Santander. Santillana itself is a bit touristy though.
• The Picos de Europa mountain range, spanning Cantabria and Asturias, is beautiful and worth at least 2-3 days of exploring. Potes is a charming base in the east (Cantabria), and Cangas de Onís is a good base in the west (Asturias) close to the sanctuary at Covadonga.

Asturias

• The coast immediately to the north and north west has some beautiful beaches west of Llanes, and a handful of pretty fishing villages (Llastres, Tazones). There are also prehistoric caves (Tito Bustillo) in Ribadesella that are worth the visit, with advance reservations required. You visit the real cave, not a replica (!).
• I never made it to Gijón so cannot comment.
• Oviedo is a pleasant city once you go past the busy ring boulevard, but it is not necessarily essential. To me the main highlight is the 8th century churches up on Mt. Naranco.
• Southwest of Oviedo, there are some wilder mountainous areas (Somiedo) that I would really like to explore someday - I haven't made it there yet!
• The coast west of Avilés is really stunning. I was just there this summer. Cudillero is super-cute, Cabo Vidio has great views, Luarca has a charming harbor. Avilés itself is where my grandfather was from; the city center is well-preserved but objectively, there are no major highlights.
• Haven't really explored the westernmost part of Asturias; I have read good things about Tapia de Casariego (coastal) and Taramundi (inland, very rural).

Galicia

• Just across the border, Cathedrals beach (Praia das Catedrais) is definitely worth a low-tide visit, with prior registration in high season (there is a visitor cap). It is busy, but impressive.
• The coast to the west did not impress me as much. I spent a few days around Viveiro this summer, but I much prefer the Asturias coast. The towns are a bit too gritty for me, and the invasive eucalyptus forest has really taken over the landscape.
• One major exception is the Herbeira cliffs. Those are truly stunning. There was too much wind for me to safely enjoy the views when I visited, but still, wow. And San Andrés de Teixedo is worth the side trip.
• Inland, the city of Lugo has impressive Roman walls and is otherwise a lovely place to visit!
• I barely remember A Coruña (visited for the day, many years ago) but my vague impression is good.
• The coast west of A Coruña has some incredible beaches, I highly recommend a side trip all the way to Cape Finisterre.

Then of course you have the rest of Galicia to explore, but that's not strictly between Bilbao & A Coruña so I will not dig into it.

Wow, Balso's reply was amazing, concise, clear and very informative. I would go by that person's recommendation....

If you do rent a car, make plans to spend a night or two at the Parador de Fuente Dé in the magnificent Picos de Europa. The drive through the gorge going to the Parador is awe inspiring and a stop at the museum Centro de Interpretación de Los Picos de Europa on route N621 is worth it. Take the cable car across the road from the Parador to the top of the mountains.

I didn't have a car, so my experience is a lot more limited than balso's. I do agree about the skippability of Santander (not a bad place, but mostly less than 100 years old due to a massive fire). I can add these as possible stops:

• Leon: Attractive historic center with lots of lively restaurants and bars, plus one building designed by Gaudi.
• Astorga: Some modernista architecture
• Betanzos: Nice not-too-large city in Galicia, not far from A Coruna (which I also liked a lot). Didn't seem to have any foreign visitors aside from me.

Also in Galicia but farther west:

• Santiago de Compostela: You don't have to be religious to find SdC lovely, though you must expect a great many pilgrims.
• Combarro: Cute but touristy small town on the west coast.
• Cambados: Attractive small town on the west coast with few, if any, foreign tourists.
• Tui: Border town with cute historic center, just north of the Minho River.
  

I think the very old churches are the major reason to go to Oviedo. It's not tiny, but it's a quiet city. I've read that the locals go to the industrial/port city of Gijon for excitement. I day tripped to the latter and found some attractive early-20th-century buildings; there didn't seem to be more than a small area of older architecture.

We were in that area with a car last fall and loved it. We especially liked Llanes and the Picos.
While we stayed at another property about 15 minutes away from the Parador, we did go to that area and took the cable car to the top and hiked around. We did day trips checking out the coastal area from Llanes as well as some checking of the mountains near there.
Just fabulous.