For transportation, Bilbao has largest airport in the region. Others are Biarritz, Pamplona, and San Sebastian (in order of decreasing size and flight options). Barcelona, Madrid, and Paris are the closest airports that get nonstop flights from the US; none are particularly close to the Basque country. If you're starting in Bordeaux (which has a large airport with flights from various European cities), it's only about 2 hours on the train to Bayonne or Biarritz, or it should be an easy drive (no direct knowledge of this). You could then see French Basque country, take a bus or train to Spanish Basque country, and fly out of Bilbao (or take a bus or train to Madrid or Barcelona and fly home from there).
A car is very helpful since there are lots of small towns and pretty countryside. Of course, you have to figure in drop fees if you pick up in France and return in Spain; again, you could take a bus or train between the countries and use two separate car rentals.
As for what to see, I'm the wrong person to ask, as it wasn't my favorite region. I'll simply say that the attraction of the region (mostly) is small towns, fancy meals, beaches, and shops (linens, chocolates, etc). If that's what you're looking for, you can have a great trip (a friend of mine loved it). It wasn't what I was looking for, or what I thought I'd find, so I didn't have a great time, except in Bilbao. I agree with Acraven that Bilbao has a lot more to offer than just the Guggenheim. However, since you are cool to Liverpool and Manchester, you may not be taken with Bilbao either. Like them, it's a former industrial city that now has only traces of its industrial past, and has several interesting museums (no Beatles connections, though).
Do look at weather for when you are thinking of going. The region is famous for its rain, clouds, and did I mention rain? Much like Ireland, there's a reason the landscape is so green. I was there in the second half of May and had great weather almost every day, but the locals made clear this was not the norm (June 1 is the start of "the season"; some places don't go on full hours until then).