From New York City

Direct: EWR → Bilbao
United

Through Madrid or Barcelona: JFK/EWR → Madrid or Barcelona → Bilbao Iberia, Air Europa, American Airlines, Delta

Through Paris or Germany: JFK → Paris/CDG or Frankfurt → Bilbao
Air France or Lufthansa

Through Lisbon: JFK/EWR → Lisbon → Bilbao
TAP Air Portugal

TIP: Consider staying a night or two in whichever city you do your layover in to extend your trip.