flying into madrid, leaving barcelona 8 days later - what should we do?

I'll leave the south alone since that should get quite a few comments.

For the north, I'd start somewhere west of Bilbao, hopefully all the way out at Coruna, and I'd haul tail to that starting point the minute I got off the plane.
I'd work east, hugging the coast then then dropping into the Ebro valley after Bilbao (note that I just skipped San Sebastián, that's personal bias).
Figure out the starting point by working backward from Barcelona (Zaragoza, Pamplona, Guernica, Bilbao, Santillana, etc - - but, obviously using your own places).

I'd give most weigh to the area west of, but not including, Santander. Second emphasis would be on Pamplona - Zaragoza. Third would be on the section just skipped. Fourth would be on Barcelona itself. Madrid would be left out entirely since it's pretty easy to incorporate into a southerly trip.

Trains are out, they all run more north/south in that part of the country. Much the same for buses, but there are some less frequent east/west ones. Which leaves a car, which is easy. If public transportation is essential, go south.

First trip to Spain, does your 8 days include travel time to and from your home?
Since you land in Madrid you may as well enjoy it. Old town madrid is fairly small and can be rushed in one full day, but there are several interesting day trips to be made from madrid. Travel to barcelona is ez via the AVE train. Barcelona at the end of may is delightful and the area is worth 4 full days.
How hectic do you want to travel?

I would divide the time between Madrid and Barcelona. In Madrid there is much to do and see. Prado is first, Park Retrio in the same area next. Followed by walking Gran Via, Puerta De Sol, Plaza Mayor,San Miguel Market, Royal Place, Cathedral Alumedna, all of which are basically a straight shot from the Park Retrio and Prado.

One day trip to Toledo should be included. AVE train from Atocha Train Station (be sure to see the tropical garden with turtles!) takes 25 minutes. When you arrive in Toledo you can take the tour bus, or better still take the local, city bus, outside the train station entrance to the Plaza Zodocover. From there walk anywhere. The Cathederal will take one or two hours to see, with the hand held audio guide. The Alcazar, now the military museum, on the high point of the city is worth a visit, as are any of the other many churches, museums, etc. El Greco's house is fascinating with many of his paintings. (Be sure to see his paintings in the the Prado as well.)
If you have access, a trip to the PradO (o not a) on the outskirts of Madrid will be interesting....the palace (actually two palaces) of Franco. Not that visited by most tourists, but gives a glimpse into his lifestyle.
Other time in Madrid to be used as you wish...great Tapas bars to be found, and much more than I have listed.
Barcelona is 3 1/2 hours from Madrid via the AVE train,which also leaves from the Atocha Station. Easy travel. Recommend the number one B&B in Barcelona, El Balcon de Born...check it out..Great location Be sure to see the Sagrada Familia, the Church of Santilla del Mar, the Las Ramblas, any of the many parks, the beach, and all the other sights!!!

8 days is not very long to fit in much more than Madrid and Barcelona. HJ's comments on Madrid and Toledo are great and remember, even if you don't like art much, the Prado is easily one of the world's greatest art museums. Try and make time for that.

The problem with heading north to Bilboa or San Sebastian is that it can take close to a day of travel. So you would lose almost two days of an 8 day trip. And while heading towards Andalusia is faster, there is so much to do there, it couldn't begin to do it justice. Instead, I'd stick to a base in Madrid and one in Barcelona and add a few daytrips if you wish.

Madrid - 2-3 days plus a daytrip to Toledo plus one to Segovia if interested in the amazing Roman aquaduct.
Barcelona 2-3 days plus a daytrip to Montserratt and/or one to Figueras.

thanks for th advice - based on your input, looks like we should split th 8 days between Madrid and Barcelona with day trips. You recommended a b&b for Barcelona - we will check that out. Does anyone have a recommendation for a charming hotel in Madrid? Thanks, Kim