If you are travelling anywhere in Spain (and you are a British citizen) you need to have your "E111" form (now a card) with you. You can obtain this from a Post office and it entitles you to the same health cover in most EU countries that you would get in the UK.
Most holiday makers also (or alternatively) opt to take out medical health insurance that additionally covers for items (normally) including emergency transport home, a stay in a private hospital etc.
Always check for dentistry treatment which is often excluded or very limited on many private health policies.
If you are planning to move to and live in Asturias, then you will pay social security in Spain and this will entitle you to national health cover.
Ensure that any health insurance that you take out covers all eventualities including the cost of transporting you (and if necessary your family) back home.