If you would rather hang out by the sea, there are a few beaches close by which can easily be reached by bus or the Metro.
Check out the website of the Bilbao tourist information which has all kinds of useful links.
You can also download an Android or iPhone app with tourism info about Basque country
In case you have a free day you should consider visiting one of the three capital cities near Bilbao: Donostia-San Sebastian(Km), Vitoria-Gasteiz (66Km) or Pamplona-Iruña(157Km). There are buses setting out to all of them from Termibus .
Apart from that here you are a selection of other closer towns and what they offer:
Portugalete-Getxo: Shared by this two towns which you can arrive by metro, you will discover the famous Hanging Bridge, wold heritage by Unesco.
Gernika: Beautifull village in the coast bombed in the civil war and where the medieval parliament of Biscay is located. You can go there with Euskotren from Atxuri. Nearby the visitor has wonderful beaches, and the remarkable painted forest of Oma. This all in the Biosphere Reserve of Urdaibai.