You probably already know about the
cathedral and the
Cámara Santa.
Here is the official website to the Naranco sites, which are IMHO, the most spectacular of what Oviedo has to offer. It’s several kilometers up from behind the train station, and there are buses if you don’t want to walk. The two buildings are about a 2 minute walk apart, and there is only one attendant. So while one is open, the other is closed. Getting there right at 9:30 (such a late hour for pilgrims anyway!) means you can see them both before buses start arriving.
San Julian de los Prados is outside of the historic core but well within the city of Oviedo (as its name suggests, it used to be out in the fields, but is now pretty well bounded by roads and buildings. You have to take a guided tour.
Since you will have several days, you’ll have plenty of time just enjoying the casco histórico, the modernist area, even the Calle Gascona where thousands of gallons of sidra are consumed every day. The parks, particularly the Parque de Invierno and San Francisco are lovely. The Parque de Invierno is a bit outside of the center, adjacent to the university, and with just amazing views back to the Picos. San Francisco is the in-town park where peacocks strut and lots of families and children enjoy. Oviedo has one of the richest traditions of public statues — Botero’s mama, a statue of Woody Allen, the locally known “el culo” are all over the center.
I think Oviedo ranks right up there with Zamora, Salamanca, Mérida, Cáceres, and all the other monumental cities the camino passes through. If you like to mix your walking with visiting, this is a great city for a few days of “touring.”