By plane
Bilbao-Bilbo: Loiu airport (http://www.aena.es/csee/S...) is the largest in Basque Country, with a stunning terminal building designed by Santiago Calatrava. It has cheap flights with Easyjet and Vueling to London and with Vueling to Barcelona and Malaga. There are many connections to other European cities with other companies as Iberia, Lufthansa, Air Berlin, Bruxelles Airlines.
Vitoria-Gasteiz: Foronda airport.
San Sebastián-Donostia: Hondarribia airport. It has flights to/ from Barcelona and Madrid.
Biarritz: it has cheap flights with Ryanair from Dublin, LondonStansted, Shannon Ireland and Frankfurt, and with Sterling Airlines from Oslo and Copenhagen.
By road
Sooner or later people normally stop. It's better to hitchhike at the exits of the towns where the traffic is calmer than on the motorways. So many long-distance truck routes cross the Basque Country it should be quite easy to find someone to take you directly from Madrid or Paris to the Basque Country, and vice versa.
By train
Daily connections to main cities from Paris, Madrid and Barcelona. Check RENFE Spain (http://www.renfe.es/) and SNCF France (http://www.sncf.com/) websites.
By car
The motorway between Paris and Madrid goes through Baiona/Bayonne, Donostia/San Sebastian and Vitoria/Gasteiz. The tolls between Baiona and Donostia are pretty expensive. Another motorway links Iruña/Pamplona with Barcelona There are motorways between all the main cities: Bilbo-Gasteiz, Bilbo-Donostia, Donostia-Iruña, Gasteiz-Iruña.