From Santiago de Compostela all the way to Barcelona is quite a distance, consider you will be traveling from the extreme west of the country all the way to the extreme east. There is so very much to see in between, I would not try to squeeze in Northern Catalonia (i.e. Figueras), as it's a bit out of the way of the route you would be taking if driving. If you would like to include Northern Catalonia, I would just fly directly from Santiago de Compostela to Barcelona, then rent a car and drive north, making a base at Girona (much better choice than Figueras).
Now if you want to make the drive, I would recommend something along these lines, once you leave Galicia heading east:
-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. From here, one can also do a nice hike to the Lakes of Covadonga, which takes about 3:30 hr each way, but are well worth it.
-Picos de Europa National Park - Mountain range between Cantabria and Asturias offers some of Europe's most spectacular and breath taking mountain scenery, definitely worth a visit. Take the Fuente Dé cable car that lifts you up high into the peaks.
-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.
-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.
-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. One can spend the night here too.
-Bilbao (2-3 nights) - 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.
-Drive along the Basque Coast - The drive from San Sebastián to Bilbao is very picturesque, with its hidden coves, seaside villages, and stunning cliffs. A highlight for me was San Juan de Gaztelugatxe, which is a 10th century Hermitage set atop a craggy islet, they filmed some scenes from Game of Thrones here.
-San Sebastián - an international culinary capital, feels very Belle Époque... maybe a bit too posh for my taste. Of course the main attraction are those famous Pintxos bars that line every street, I feel they are a tad overated.
-Olite - walled medieval town dominated by the Royal Palace of Olite was the seat of the Kings of Navarre. For me, it's one of the more impressive medieval palaces in Europe.
-Zaragoza (2-3 nights) - the capital of the neighboring region of Aragon. The city has more than 2,000 years of history including Roman ruins of the city walls, forum, and amphitheater, as well as the Aljafería; an 11th-century Moorish palace. The best part, Zaragoza is relatively undiscovered as far as major Spanish cities go.
-Poblet Royal Monastery - a UNESCO World Heritage Site and still active 12th-century Cistercian monastery. It was a fortified medieval royal residence and contains the unique hanging tombs of the old Kings of Aragon.
-Tarragona - ancient seaside town with very good Roman ruins, including an amphitheater, intact aqueduct, and Praetorium Tower, was once the capital of Roman Hispania (modern-day Spain). Tarragona's medieval old town is a delight to explore too.
Hope this all helps :)