By train

It is not possible to travel within one day.

There are two options for the direct route:  

1. The fastest: from Toulouse to Irun with french SNCF, and then from Irun to Oviedo with Spanish RENFE. (To plan that travel, you might also use the web site of the Deutsche Bahn , that combines information from SNCF and RENFE.)
2. Perhaps more touristic (in any case more difficult to get information): from Toulouse to Hendaye with french SNCF, from Hendaye to San Sebastian with the "Topo" (via Irun, several trains per hour), from San Sebastian to Bilbao with Spanish EuskoTren (every hour; 1st departure 7h47, arriving 10h22; last departure 19h47, arriving 21h22), and from Bilbao to Oviedo with FEVE (via Santander, e.g. departure 13h35, arrival 20h41).

By plane

The only direct flights between Toulouse Blagnac airport and Spanish mainland go to Madrid (7 daily flights). Toulouse being well connected to main European cities, you might get to Oviedo airport via other ways. You might also choose to fly to Bilbao and join Oviedo by train.