By sea;
Santander is the nearest ferry port to the Picos de Europa. Brittany Ferries sail from Plymouth and Portsmouth to Santander as well as running a route from Portsmouth to Bilbao.
From Ireland there's a ferry crossing from Rosslare to Bilbao.
Add interest to your crossings by looking out for whales, dolphins and seabirds. Visit the Company of Whales and Orca websites for information on the possible cetaceans you might see in the Bay of Biscay.
By air;
Santander also has the closest airport to the Picos de Europa. Ryanair currently fly from the U.K. (London Stansted, Edinburgh, Manchester or Birmingham) and from Ireland (Dublin) into Santander. For Belgian and Dutch visitors there are also flights with the same company from Brussels to Santander.
Easyjet fly from London Gatwick to Bilbao in the Basque country and now also from Manchester to Bilbao.
Vueling fly from London Gatwick to Santander, from Birmingham to Oviedo and from Manchester to Bilbao or Oviedo.
By land;
From the U.K. - A more environmentally-friendly way to travel to Spain if you want to avoid flying. Information from the man who knows all about European train travel, the man in Seat 61, rail to San Sebastian/Donostia. From there you can get a bus to Bilbao where you can be collected.
From Madrid - the most direct route to drive is via Segovia, Valladolid, Palencia, Aguilar de Campoo, Cervera de Pisuerga, over the Piedraslenguas pass bringing you down to Potes in the east of the Picos.
From France - from Bordeaux/Biarritz, crossing the border at Irún in the Basque country, take the A8 motorway past Santander and Torrelavega to Unquera, then turn onto the N 621 for Panes. Here you can head either to Potes and the eastern Picos, or to Cabrales and Cangas de Onis and the western Picos.