Personally I would stick to northern Spain. San Sebastian is pretty much the undisputed capital of food in Spain - not cheap but extraordinary. So that or Bilbao are obvious choices.
However, if it were me I'd do the following. I should say that I run an Anglo/Spanish travel company so there's some some background to this. Start in Santiago de Compostela in the far northwest of the country. Fantastic seafoods and wines in the Rias Baixas area.
Continue through some of the most remote and untouched mountain/rural scenery in Spain. Oscos mountains and then into the Picos de Europa which have already been mentioned. You are essentially travelling through mountains close to the sea so on the one hand you have really good game and meats, on the other fish and seafood.
From the Picos I might head to Bilbao (it's about 3 hours' driving) at least to visit the Guggenheim. Their set lunch is very good by the way.
From there I personally would not go to San Sebastian but instead drop an hour south into La Rioja. Three things to do here, eat in Casa Toni near Haro. Stunning food, really well done. Go the central market in Logrono to look at the quality of the raw materials these guys use. There is a wine shop on the corner in front of the market which, as of October this year, had some red wine for 2 euros a bottle. No label. In fact it was a semi crianza and fantastic - well it would have been a first rate bottle for 10 euros, at 2 it was fantastic. Carry onto Calle Laurel which is famous for its tapas - follow your nose and mouth.
Finally, and this is where I would end my trip, in a small village in the south of Rioja is a restaurant called Venta de Moncalvillo. They offer a 'maridage' menu whereby they serve a distinct and distinctive wine with each course of the meal (there are at least 6 courses). The place is run by 2 brothers - one a self taught but now very well known somelier. The other a self taught chef. This place is one of the best places to eat that I know of - not pretentious, just brilliant, imaginative food.
So, I hope you're still awake but that would be my opening gambit. This way you have seen some of the most significant cultural regions of Spain (travelling along the Camino de Santiago), certainly its most beautiful scenery and you have a great variety of foods and wines as you travel too. May should be gorgeous - may be rainy but there will be snow on the top of peaks, flowers in the meadows, everything green. Perfect.