DH and I went about 5 years ago without DC. We like hiking and the best hikes we did were small sections of the Camino del Norte.
We travelled around on local trains and buses but they were quite unreliable, sometimes not turning up and lots of replacement buses instead of trains. I would hire a car.
The best places we stayed were more remote, outside the towns.
This place was lovely, had great food and good access to the Picos de Europa: www.tripadvisor.co.uk/Hotel_Review-g670808-d573880-Reviews-Hotel_La_Casa_de_Juansabeli-Arenas_de_Cabrales_Cabrales_Municipality_Asturias.html
This place in Ribadasella was nice, but there weren't many places to eat nearby, although you could walk into the town if you didn't mind a long uphill trek. Ribadasella had good access to coastal walks, which we did in both directions on 2 different days. www.tripadvisor.co.uk/Hotel_Review-g660742-d2483052-Reviews-La_Biesca_Sebrenu-Ribadesella_Asturias.html
We also had some time in Cangas de Onis, which was quite busy and you need to get special national park buses to access the Picos as vehicle access is limited.
We also stayed in Llanes, Santander and Bilbao. Llanes was nice and we did some good coastal walks from there. Santander was less impressive and the main thing we liked about Bilbao was the Guggenheim.
It was August, school holidays. Weather was mixed, some sun, some rain, but not really hot like other parts of Europe and good for hiking.
We felt we wouldn't have wanted to just stay in one place as a base and the good things to do were quite far apart and even with a car would have involved a lot of driving. We found after a couple of days out in one place we were happy to move on as we'd done everything there.
But others might have better recommendations about a good place to base yourself.