Bilbao Airport (BIO) is the main airport of the province of Biscay and it is situated 9 kilometres from the city of Bilbao. Being the tenth biggest city in Spain and the capital of the province, Bilbao’s popularity increases over the years, including the airport which annually has more than 5...
This is what you are looking for (copied from that website):
There are no direct railway links from the airport to the city, however if you need to get to the main train station Estación de Abando, take the previously mentioned bus line nr.3247 and exit at the bus station which is located 2 mins walking distance from the train station.
Peaceable Projects Inc. is a U.S.-based non-profit group that brings the vast resources of the wide world together with the ongoing needs of the people who live, work, and travel on the Camino de Santiago pilgrim trail network in Spain.
It is not far. The regular airport bus referred to above leaves every 20 minutes or so from the airport. There are stops once you reach the city, and the train station is near the first stop. I am unclear from the information above what is meant when they say the bus station is two minutes walking from the train station. The train station is at one end of the main street, and the Termibus, ie bus station, to all intents and purposes is at the other end, Maybe the metro, located adjacent to the Termibus, links with the train station? Obviously, a taxi will solve many problems.... but depending on the luggage involved, it is much more cost effective to take the airport bus and sniff your way to the train station. The bus drivers will know where to let you off if you ask for Abando.
If you do not have much luggage, I suggest you take the bus A3247 and get down at the second stop in Bilbao (Gran Via 46 - Sota). From there, follow the Gran Via street in the direction of the old part of Bilbao (Casco Viejo) and the train station at Abando is only like 10 min away by foot. Its an easy straight line walk.
The taxi from the airport will cost you around 30 Euros and the current bus fare is 3 Euros.
The bus ride takes about 20 min to reach Bilbao.
The bus timings can be found here (the website is in Spanish - look for A3247): https://web.bizkaia.eus/es/web/bizkaibus/lineas
If you have more than an hour and a half for the transfer, going by bus is the best and cheap option. As @David mentioned, the bus stop is at the very end of the airport and the tickets can be bought before you board the bus and timings are displayed there.