here are some hints that most tourist dont know and are a must.
in front of the Guggenheim museum, other side of the river, take the funicular train up the hill (Funicular de Artxanda). Once up, turn left for 50mts for a breathtaking view of Bilbao (it it��s not raining). Very cheap and much worth it. Good restaurants up there.
Take the metro to Las Arenas /Areeta (everything is written in spanish and Basque), 20min ride, follow directions to Puente Colgante (Hanging Bridge), unique in the world, take the elevator upstairs for a breathtaking view of the Bilbao Estuary. You are in Getxo, plenty of beaches and good restaurants, posh town near Bilbao, for excellent fish go to the old port (Puerto Viejo) at the end of Ereaga beach, up the stairs, beautiful surroundings. If you want to eat in an old windmill, ask for Aixerrota (in Basque, windmill) restaurant (may need a car).
Try to visit and eat at a Sidreria, set menu for around 25 to 30 euros, fresh cod, cod omelette, t-bone steak on charcoal, cheese and walnuts and all the cider you can drink (this is natural cider, not sparkling and no sugar, low alcoholic). Very typical places, most common in rural areas, there are also a few in Bilbao, popular atmosphere.
At the old town, dont leave without eating pintxos (similar to tapas, small elaborated appetizers) very good ones at Cafe Bilbao or Victor or Victor Montes, at the Plaza Nueva (heart of old town, not to miss). Help yourself with your own hand to as many as you want and pay later, trust is the word.
Once i get your reply with more info about your visit, I may be able to recommend other restaurants.