Flying to Oviedo.
The local airport is actually not in Oviedo but in Ranón, a 30-minute drive from it. Bus company ALSA has regular buses from the airport to Oviedo and a taxi may take you for around 20 euro.
Several national airlines (Iberia, Spanair, AirEuropa) operate from Ranón airport and some low-cost international airlines (Ryanair) have been added recently. A Madrid-Oviedo ticket might go for 40 euro (discounted).
Another way to get here is by bus. Again, ALSA is the company that operates most routes between Oviedo and other Spanish cities. Coming from Madrid, the ride is 5 hours and costs 27 euro. The buses are quite comfortable. However, some people like trains better: RENFE also operates between Oviedo and most major Spanish cities. The trip from Madrid is a little longer than by bus and a little more expensive.
If you're driving your own car from the South, beware that there is an expensive toll right before getting into Asturias, if you take the Huerna highway. There are alternative local roads, you can check www.mappy.com for all sorts of directions.
Although the best way to get around Oviedo is on foot, local buses will take you to most places for 85 cents in the likely event of rain or if your feet are sore. Buses are quite new, comfortable and air-conditioned. You can pay per ride or purchase a bus card with any amount of rides that will lower the cost per ride, at any newstand.
Taxis are an alternative if you are going a little ways, or if you're going back home at night and don't feel like walking because you had one too many culines of cider. Taxi fares start at 2,90 euros and you may go anywhere in the city for less than 6. Tips are not expected.
Actually, the only way of getting around Oviedo that I do not recommend is driving your own car. Parking is limited and expensive, and most streets in the historical part are only open to bus and taxi.
To get to Oviedo if you prefer public transportation (I do!) you have several choices. The bus, which is the most expensive, the FEVE train or the RENFE Cercanias commuter train. The RENFE trains are regular, reliable, clean and cheap, cheap, CHEAP! You can enjoy the landscape more if you take the train. The train station drops you about 1/2 to 1 mile from where the "action" is in Oviedo, but it's close enough to not be a bother. It's very easy to get to center city from the train station- you just walk down the long boulevard and before you know it you are there.
God gave you these two things on the end of your legs called feet-- this is a good place to use them. Oviedo has great public transpo, but for most everywhere you want to be, you will want to walk. There is too much great stuff that you'll miss if you're in a vehicle.
Alsa is the national bus company, they run a comprehensive service throuhout Asturias. The journey from the airport takes around 40 -50 minutes.