YY to the Picos de Europa. We had a wonderful week hiking around them a few years ago (just me and DH mind, never done it with kids). It's beautiful. We stayed in Potes which is a lovely town and pretty central.
There's a scenic little railway from Bilbao to Santander called the feve. Here's a Simon Calder article on it. You could stop at a few places en route from Bilbao (love the city) to Galicia and pop into the Picos.