I was in the Basque country from May 12 through 22, 2017. I'll start with the logistics to help others plan, and then get to the trip itself.
First, my itinerary:
May 11, Thursday: Fly JFK to Madrid on Iberia
May 12, Friday: Fly Madrid to Biarritz on Iberia (don't worry - same ticket); overnight in Bayonne, France
May 13, Saturday: Day trips to St. Jean de Luz and Biarritz; overnight in Bayonne
May 14, Sunday: See Bayonne; overnight Bayonne
May 15, Monday: Train from Bayonne to San Sebastián, Spain; overnight in San Sebastián
May 16, Tuesday: Day trip to Hondarribia; overnight in San Sebastián
May 17, Wednesday: Day trip to Getaria: overnight in San Sebastián
May 18, Thursday: Bus to Vitoria-Gasteiz; overnight in Vitoria
May 19, Friday: Bus to Bilbao; overnight in Bilbao
May 20, Saturday: See Bilbao; overnight in Bilbao
May 21, Sunday: See Bilbao and Portugalete; overnight in Bilbao
May 22, Monday: Fly Bilbao to Madrid and Madrid to JFK on Iberia
Flights were a bit tricky. I initially thought of going in reverse order, but it turns out museums are a big part of seeing Bilbao and Vitoria, and starting in Bilbao would put me in these cities on a Monday when the museums are closed. I considered flying to San Sebastián rather than Bayonne because it would have saved me about $150, but I didn't want the extra hassle and exhaustion of taking a long bus or train ride after two flights. So, I paid $788 (including extra fees for seat selection - $33 each way) on Iberia for the flights above (all on one ticket, of course).
The plane from Madrid to Biarritz was VERY small (still a jet). So small, that anything larger than a personal item had to be gate checked. They gave us tags in the waiting area, and as we walked on the tarmac to board the plane, there was a rack to put the bags. They were all waiting there when we deplaned an hour later. But I mention this as a reminder to always have the most valuable items ready to be pulled out of any luggage, so even if you have to check a bag unexpectedly, you can keep things like medications in carry on.
A note: Lonely Planet France (at least in the 2009 edition I was using) called my arrival airport Bayonne-Anglet-Biarritz (BAB). However, when searching for flights, you won't find it under either that name or those initials; it doesn't even show up as the Bayonne airport. It's the Biarritz airport, code BIQ.
My accommodations were quite a variety: an apartment hotel, a small pension, an inexpensive medium sized hotel, and a relatively fancy large hotel (fancy for me, that is).