There is a bus line that connects Sondica Airport in Bilbao with Bilbao city center (Plaza Moyúa, near the Carlton Hotel).
This bus line also stops on the way at Parque Etxebarría and Calle Enrique Ibarrieta. The ride is about 30 minutes.
If you wish to continue your trip to Burgos by bus, you must go to the Termibús Bus Station. There you will find the company ANSA, which operates buses to Burgos. It is located on calle Gurtubay, near the San Mamés football stadium.
In order to get there you can take a taxi or the subway. If you decide to take the subway
you should take Line 1 to Plentzia and get off at "San Mamés" (2 subway stops).
If you wish to continue your trip to Burgos by train, you need to go to the Abando Train Station on the Plaza Circular. If you do not have a lot of luggage you can easily walk from Plaza Moyúa. It is a pleasant 10-minute walk:
Plaza Moyúa >> walk down the Gran Vía >> Plaza Circular
However, if you have too much luggage to walk, it is advisable to take a taxi or the subway. You should take line 1 to Bolueta and stop at "Abando" (1 subway stop)