All public transport in Bilbao is very cheap, clean and regular.
The bus from the airport runs every 30 minutes, costs 1.30 euros to the bus station (termibus) and you pay on the bus. It will cost you between 25-30 euros in a taxi.
On the regular buses you can get single or return tickets which you purchase from the individual bus stops. Tickets have to be validated there before you get on the bus. It's easy to understand after the first go.
On the metro it is exactly the same, the machines have all the destinations and it is a touch screen process in various languages. The longest trip we could take on the metro only cost 3.40 euros return. Tickets have to be validated at the same machines.
You can purchase a Bilbao Card from various places which covers all public transport in Bilbao itself but you will need to pay more for trips further out. The cards can be bought for a single day but there are other time periods.

www.termibus.es/index.php?idi=en
Google map: bit.ly/tOrwmQ

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