Welcome and congrats for your plan. Not knowing your budget, here some thoughts:
• Local flights, trains and car rentals (rentalcars.com, etc) are very affordable
• Look into the possibility of flying from home directly into Madrid and back from Barcelona.
• After seeing the sights in Madrid, take a train to Pamplona or Logroño, start the camino and and reach SdC in about 3-4 weeks
• In SdC, buy some summer clothes, rent a car and take a leisurely drive: Finisterre > Muxia > A Coruñia > Gijon > Santander > Bilbao, about 4-5 days. (Leave the highway after Gijon to see places like Llastres, Ribadesella, Llanes, and furter down the lovely village of Santillana del mar!) Book accomodations on the go (Booking.com, etc)
• Take a flight from Bilbao to fabulous Barcelona* before leaving back home.
(*) As you will have enough time, you might look at the option of flying first to Palma de Mallorca for a couple of days, (Air Europa at less than 80 Euros)
You'll have an unforgetable Spanish experience without rushing!
Buen camino