Avilés is the nearest city to Asturias airport and is about 10 minutes away. It has a history as a port and used to be home to a lot of heavy industry. However it has been cleaned up dramatically and now is a great place to stop off on your way to or from Granero del Burro. It has the most beautiful historic centre and you can book the NH Hotel in the centre which has its front doors opening into the main square.
Everywhere is accessible of foot and it is worth exploring the network of historic streets and bars. Cider is the Asturian speciality!! There is a fantastic restaurant called Terra Astur. Don't plan on eating anything else for the day if you have lunch here!! Reading the menu is like reading a novel - it is huge!!
In addition to the old town there is the Niemayer Centre down in the port area. This is a modern cultural complex designed by the Brazilian architect Oscar Niemayer and is well worth a visit. You get some great views of the port as well.
Aviles also has its own beach. Salinas beach is about 10 minutes walk from the main square or 5 minutes in the car. Once again it is well worth a visit at any time of the year.