Make sure you visit the Picos de Europa National Park
Within the city, there are various museums, historical buildings, etc. Be sure to walk around the centro antiguo, that being the part of Oviedo that was once encircled by a stone wall. The Cathedral, convent, and parts of the wall itself are just a few things to see. The Parque San Francisco is one of the biggest parks in Oviedo, complete with gardens, paths, ponds, and pavos reales peacocks. This is just off of Calle UrÃa, one of the main shopping areas down town. El Museo de Bellas Artes, in the plaza of the Cathedral, contains works of Asturian artists as well as Picasso and El Greco. To get away from the city, try a walk up to El Cristo, located on El Naranco, which is visible from nearly all parts of the city. On the way up are two pre-romanesque buildings. La Pista Finlandesa is another nice option for walks or runs, as it skirts the Naranco hill and has full views of the city. Also, keep an eye out for the numerous statues and sculptures in Oviedo.
Night life in Oviedo is great. Try the Calle Mon, located just off of the Cathedral's plaza, but remember that most Spaniards don't leave home before 12:00. If you're an early bird, head to the Calle Gascona, just north of the Cathedral, which is said to be home of the largest number of sidrerias in the world. Sidra is an alcoholic apple cider famous to Asturias, with its own tradition. Be sure to ask a server how to correctly drink the beverage, and don't miss the way they pour it. Another game that can be played in many of the Bars in Oviedo and other parts of Spain and France is Duro. It is a drinking game similar to that of quarters. They play it with a drink known as Calimocho, a combination of Coke, red wine, and current berry syrup. Ask a bartender and they will give you the necessary cups, etc.
But Asturias is not only Oviedo. Try visiting coasts cities like Gijon and Avilés, spend a morning in Tito Bustillo caves, or relax in one of the thousand beachs in the province.