We will be leaving the Rioja wine country on a Sunday and were initially planning to spend a full day (Monday) in Bilbao before moving on to San Sebastián on Tuesday. However, the Guggenheim museum is closed on Mondays and we are now wondering if we should visit another town other than Bilbao, or keep to our original itinerary. We would be grateful for any ideas and recommendations. Thanks in advance!
Bilbao has a rather large and attractive historic district, the Casco Viejo. If the Guggenheim is really important to you, you could go straight to San Sebastian late Sunday and then day-trip to Bilbao on Tuesday. There's frequent bus service, so you wouldn't necessarily need to drive.
Alternatively, you could go to Vitoria-Gasteiz, the capital of the Basque Country. It has a large medieval district on the top of a hill with outdoor escalators to assist in getting to the top. There's a playing-card museum, an old cathedral (under renovation when I was there in 2016; I don't know the current status), and a couple of modern-art museums.
From June 17 to September 22 the Guggenheim opens on Mondays, when are you going? Also, Bilbao has lots to enjoy besides the Guggenheim, it´s a city with stunning architecture, a funicular, Azkuna Zentroa, the funicular of Artxanda and a beautiful old quarter, as well as a really nice 19th century area full of great architecture. On the outskirts (Getxo and Portugalete), the unique Puente Bizkaia, a UNESCO World Heritage Monument.
If you’re driving you could head east, then north, and visit any number of smaller towns and sites such as, Estella, Puente la Reina, Monastery of Iranzu, Alto de Perdón, or even Pamplona.
If heading towards Vitoria-Gasteiz you could make a quick stop at Ermita de la Purísima Concepción.
Santander and the towns near it are nice.