I feel northern Spain is best experienced with a car, which allows you to stop at the more off the beaten path sites like Loarre Castle or the Olite Royal Palace.
Here would be some of my top picks for cities and day trips (East to West). I have bold places to spend some nights and later radiate out to explore the region. Day trips are un-bold:
Zaragoza (2-3 nights) - Capital of the Aragon region. More than 2,000 years of history including Roman ruins of the city walls and amphitheater and the Aljafería, an 11th-century Moorish palace. It also has the famous pilgrimage site of Nuestra Señora del Pilar basilica, housing the miraculous image of the Virgin Mary.
Royal Palace of Olite - For me, one of the most impressive medieval castles in Europe. The seat of the Kings of Navarre. There is a also pretty good Parador hotel next door.
Loarre Castle - One of the oldest castles in Spain, it was featured in Ridley Scott's epic film Kingdom of Heaven. The Loarre Castle is very picturesque as it sits high on a cliff overlooking the southern foothills of the Pyrenees.
San Sebastián (2 nights) - an International culinary capital, it's elegant art nouveau buildings flank its picturesque sandy seashell beach. Feel very Belle Époque... maybe a bit too posh. Of course the main attraction are those famous Pintxos bars that line every street.
Bilbao (2 nights) - Winner of the 2018 European City of the Year, Bilbao is the more grounded brother to San Sebastián. With its contrasting old and new architecture and nice riverside area, I actually like Bilbao better than San Sebastián. The main draw is the Guggenheim, one of the leading contemporary art museums in the world.
Santillana del Mar - Well preserved medieval town in the Cantabrian region. Known for its medieval towers, Renaissance villas and the Romanesque Santa Juliana Collegiate Church.
Altamira caves - UNESCO world heritage site located near Santillana del Mar with numerous Paleolithic cave paintings. The real caves are closed to the public but there is a pretty good replica cave and museum there too.
Potes - (2 nights) an atmospheric medieval town located high up in the Cantabrian Mountains. Makes for a great place to explore the Picos de Europa National Park in depth.
Picos de Europa - Mountain range between Cantabria and Asturias offers some of Europe's most spectacular and breath taking mountain scenery.
Oviedo (2 nights) - The capital of Asturias. Known for its Gothic medieval old town, apparently Woody Allen's favorite city in Spain.
Covadonga - known as the "cradle of Spain", a pilgrimage site dedicated to the Battle of Covadonga of 718 AD which set off the Reconquista. There is also a Holy Cave, where lies Kings Pelagius and Alfonso I, the first kings of Asturias.
Hopefully these give you a few ideas :)