(Almost) Completely Unplanned Road Trip

Northern Spain is quite large---you could get fairly extensive in just the Basque region for example, so I would read up to see what you want to rule out before getting too far.
I'd also add what month, as that can give you data on what to exclude (San Sebastian during the film festival, etc.)
You'll be having to connect in Madrid in all likelihood, but I might dig into the logistic on the flights to see how that affects things.
I would also recommend Booking.com -- I don;t know about Expedia, bit Booking is good about having small hotels and all types of accommodations, which might be helpful for small towns.

How about if I stuck to Asturias? I'm just in the spitballing stage right now.

Time of year?

Keep in mind that on a US road trip you may be traveling through a lot of fairly uninteresting territory with not much to see. That is not the case in northern Spain, or in Europe in general. Therefore, you'll need more time if you're unwilling to blast past a lot of interesting places, such as:

Barcelona
Puigcerda
La Seu d'Urgell
Hondarribia
San Sebastian (probably should prebook in peak season)
Bilbao
Vitoria-Gasteiz
Laguardia
Burgos
Picos de Europa
Leon
Astorga
Oviedo
Betanzos
A Coruna
Santiago de Compostela (gets lots of pilgrims; might need to prebook)
Pontevedra

Those are some of the places I liked when I covered that area by public transportation. It took much, much longer than 3 weeks. There are lots of interesting small places along the northern and northwestern coasts I haven't listed, and they'd be easier for you to reach with a car.

Edited to add: Limiting your target area to Asturias or mainly Asturias would allow for a relatively relaxing trip, I think.

I am not equipped to plan an itinerary, just offering some thoughts to ponder. My experience is limited to Basque region, I spent ~11 days there.
I have taken baby steps at planning a potential Asturias trip -- meandering along that coast looks dreamy.
Time of year is important in the north--I'd want to avoid torrential rain if possible, even more than the heat of the south.
I do find sorting out the transport logistics can really help shape a trip though--if not flying open jaw, then you could fly to Madrid, take train to Oviedo and start there, doing a good loop.
I'd also add how much national park exploring you like to do, as that could eat a lot of time and easily rules out if you are not hikers, etc.

I've been many places in Europe but I have fallen in love with Spain. So far, we have been to Barcelona, Madrid, and Seville, two or three weeks in each place, we loved all those places, and we are planning a trip to Valencia which will also be for a few weeks duration. We don't jump around from place to place. We have time and resources. In the past we have just found a great place and lived there for a couple of weeks.

So it seems like a road trip would be something very different, and I'd like to see some new places. Oviedo seems interesting, as does Aviles. I just started thinking about it this morning. I'm just looking for ideas and advice.

As already mentioned, northern Spain is large. In 2017 we did a northern Spain trip starting and ending in Madrid. The route we took was Madrid-Zamora-León-Santiago de Compostela-Ribadeo-Santillana Del Mar-Hondaribbia-Bayonne-Pamplona-Santo Domingo de la Calzada-Burgos-Tordesillas-Madrid. We made plenty of stops between those lodging locations.

What we didn’t see on that trip was the northern part of Spain along the Pyrenees from Pamplona over to the Barcelona area. There are enough things to see in that area for a separate month long trip.

I wouldn’t limit myself to just Asturias, but there is plenty to see in it.

We spent almost three weeks in Asturias a few years ago and I highly recommend that region; it has mountains within a half-hour drive of beautiful coastal towns. Look at Ribasedella, especially.

See if you can find a hotel within an Indiano mansion:

https://www.turismoasturias.es/en/casonas-asturianas

From our hotel west of Oviedo we took a drive into Leon, very do-able. There is a Gaudi palace in Astorga.

Sounds awesome!
Asturias, Cantabria and Basque Country for 3 weeks.
Or even longer…

Take a look a Soria, province.

An idea for a few days might be:
Start in Burgos, head south, visit Covarrubias. You can visit the Sad Hill Cemetery route in the area, which is a group of several places used in the movie The Good, The Bad and The Ugly. Other things to visit Monastery of Silos, and castle of Ucero.

If you go slow stay but the Natural Park of Canyon of Rio Lobo. Burgos de Osma is also a nice place to stay.

Keep going on to Soria. Some nice things to see there, and eat some Torreznos.

After Soria, go east to Agreda, Tarazona, and on to Olite.

From Olite, you can swing back towards Burgos, visiting west Navarra and La Rioja. Or you can head north to San Sebastian and then go west.

There is a lot to visit. Do you prefer nature or historical places? Do Roman things interest you? Clunia and Numantia are very interesting and not many tourist.

One last question, when are you planning to go? Heat or cold could be a factor. Also you could end up visiting some different fiestas in small towns for an authentic expericence.

dlindstrom, I always plan around the weather, and surprisingly (to me at least) the Asturias region is quite cool, with the warmest months being in the middle of summer, with average highs in the low to mid-70s. That makes a mid-summer trip do-able.

Asturias doesn't extend very far south of the coast; the coastal area stays mostly cool and cloudy through the summer.

A mid-summer trip would be great, but do keep in mind the popularity of tmany of the coastal towns with domestic tourists. I think I mentioned Ribasedella, it's a lovely and interesting town, but I would try to book a bit ahead of time, unless you are willing to take "just any" accommodation.

Let me know if you want any restaurant tips for Asturias or Galicia.

Astorga (Leon region) is notable not only for the Gaudi palace, but for the particular type of food of the area, much of this derived from the Maregata (sp??) people. Within a quick drive is EL CAPRICHO, a unique restaurant where you can sit inside a cave and enjoy the steak and other meat products that some say is among the best in Spain.