BETA PHOTO: Pablo Perez-Zapardiel, NASA Madrid Deep Space Comm...
Beautiful and spectacular limestone mountain area on the north coast of Spain. Has a distinctly Celtic feel to it, architecturally, and culturally. The compact but spectacular mountain range of Picos de Europa lies in Asturias, and contains some wonderfully scenic hikes, and spectacular climbs.
The coastal and inland towns are, as is usual for Spain, charming and historic, and also as usual the food and drink is terrific, especially the seafood. Be sure to drink lots of the Sidra Natural - local hard cider which involves the ceremonial pouring of just a little into a very thin glass from a great height.
If you want to avoid the crowds visit in late June, the hotel rates are cheaper too.
From the US to Barcelona, then rent a car. I've only driven from Madrid because that's where I work some of the year, it's a fairly long drive, but the roads are fast and smooth.