We have 10 days to spend in Spain in early May. We'd like a mix of serious hiking, some cultural sites (like Bilbao) and would like to stay in nice accomodations. Any suggestions would be appreciated. Thanks.

Great hiking in the Picos d'europa which is up by Bilbao, in the area you will also find the Rioja wine region.

Thanks.

Another option would be the Poqueira Gorge or elsewhere in/around Las Alpujarras.

May could be too early to get right up into the higher peaks of the Picos or Pyrénées.
I also happen to think you should tell us more about your plans. Like where do your enter and leave Spain, method of transport too. It helps us narrow down an area for you.
Here is a website with loads of routes on them:
http://www.wikiloc.com/trails/outdoor/spain

We are fairly flexible. We will begin and end our trip in Paris, spending a few days on either side of our trip to Spain there. We can take a train, or rent a car, or a combination of both, but we want to spend about 10 days in Spain and would like to do some good hiking, see Bilboa, and stay in nice places. I hope this helps.