Aviles isn't on the "RENFE" Spanish rail network, but you can take a train from Aviles to Gijon, journey about 45mn - see
http://www4.feve.es/horarios/
from Gijon you can travel by train via Leon, Irun/Hendaye, Les Aubrais (Orleans), Nantes and Redon, with a journey time of just under 26 hours (dep Gijon 10h25, arr Lorient 12h20 the next day) Journey opportunities can be checked using Deutsche Bahn's all-Europe journey planner,
http://reiseauskunft....t=yes&protocol=http:&
An alternative might be to take the bus along the coast from Aviles to the French border, dep Aviles 13h45, journey under 9 hours, but sadly this misses a useful train connection at Hendaye leaving you with an enforced overnight stop.
http://www.alsa.es/portal/site/Alsa
Asturia airport is not far from Aviles, Gijon or Oviedo, and there is a flight from there to Paris, with journey opportunities onward from there.