Public transport, and in particular the metro, are extremely reliable and allow you to move around Bilbao with ease.
Taxis, however comfortable, are quite expensive.
For the center, and along the main points of interest, you can easily move on foot. From Casco Viejo to the Guggenheim Museum, for example, you have to travel a distance of about 1800 meters; you can walk in less than half an hour.
Here are some valuable information for getting around Bilbao.

AUTO AND TAXI

Traveling by car, for many, is a real nightmare, with unique senses, traffic, and ongoing work in progress. Better to leave the car in one of the city’s many parking lots, and then continue on foot.
Alternatively, you can use the taxis, which are comfortable but quite expensive.

METRO

Quick. Reliable. Efficient. The metro is the most practical and fastest solution for getting around the city, center and suburbs. Three lines are active (L1, L2 and L3), every day, from Monday to Sunday, including holidays.
Lines L1 and L2, from Etxebarri to San Ignazio, share the same route, before separating. The Zazpikaleak-Caso Viejo station is an interchange hub for the three metro lines.
Line L1 connects Etxebarri with Pletzia. The L2 line, on the other hand, runs from Basauri to Kabiezes. Finally, the L3 line runs from Kukullaga to Matiko. With only 7 stops, Line L3 is the shortest of the three.

Lines L1 and L2, are active every day from around 06:00 to around 23:00. The frequencies vary from 5 to 10 minutes on weekdays, up to a maximum of 20’ on Saturdays and Sundays.

FUNICULAR OF ARTXANDA

Over 100 years of history, for one of the most fascinating and suggestive public transports. In operation every day of the year, it is active in winter from 07:15 in the morning, until 22:00; in summer, from 08:15 to 23:00.
The funicular connects the lower part with the summit of Mount Artxanda. It is worth taking a look especially for the splendid view it offers over the city. More info from this link.

BUS

Bilbobus manages the local bus transport with 35 daytime lines, and 8 night lines. Buses run every day, including Sundays and holidays, from around morning until about 22:30.
BizkaiBus, on the other hand, is the company that manages long-distance connections, connecting the external areas of the city, such as Bilbao’s international airport.

GETTING AROUND BILBAO WITH THE PUBLIC MEANS OF NIGHT

Friday, Saturday and holidays, there is a convenient night service for lines L1 and L2.
Friday, and holidays, there are rides every 30′, from about 10:30 pm until 02:00 am at night. On Saturday, however, the two metro lines run all night, with rides every 15 and 30′, until 06:00 in the morning.
On Saturday, the L3 line also does a non-stop service for 24 hours, until 06:00 in the morning the following day.

To reach the areas not covered by the metro, you can use the night buses. Like the metro, they are active only on Fridays, Saturdays and holidays, with a non-stop service, every 30 minutes, on Saturdays. You can choose between the 8 night bus lines, identified by the letter G and the progressive number from 1 to 8.