When you land in Bilbao (BIO) airport you can take a cab to the city for about 15/20 euros (relatively cheap compared to most european destinations), but there is also a shuttle service to the city center for just 1 euro. The bus runs from 6 am to 10 pm.
It is very new airport, beutifull and near the city, 9 km.
There are good comunications with the city, by bus every 30 minutes during the week and every 1 hour at weekends. 2€.
There are also taxis but those are more expensive, 25€.
There is not underground/train.
We got off at the bus station & had to take another bus to the Ayuntamiento, to take a third one to the airport. We almost didn't make it, as buses pass every 30 minutes...