In Asturias, you can use any form of transportation to get around. In the city, it's very common to either have your own car, use the public buses or ride your bike. If you want to get anywhere else in Spain, long-distance buses, trains and even airplanes are a great way to travel.
The main form of transportation in Asturias is private car. The public long-distance buses (run by the Spanish company Alsa) make a lot of stops along the way to their destinations, and the trains are slow because of the mountains and very tight spaces they have to go through. So, having your own car is very helpful in Asturias. Plus, the time it takes to get to and from the destination are shorter than if you had taken the bus or train there.
In Oviedo, though, people can pick what works best for them. For example, you can take the public buses and get to your destination within about 30 minutes, anywhere in the city. Some people will use their bicycles and cars, but because of the traffic at times, many of the people will just walk. Walking seems to be the most popular form of transportation in the city center because parking is hard to find and also the number of pedestrians makes it rather hard to navigate by car.