You can qualify for state health care:
as a UK national in receipt of a UK State Pension (if one of a couple qualifies their spouse can be included as a dependent)
if you work here and contribute to the state system
emergency treatment under the EHIC scheme (that's intended to be temporary and is for a limited period (i.e. during holidays or temporary work assignments) but some people 'cheat' the system and, even as permanent residents, get away with using the EHIC card)
Apart from that you need private medical insurance.
We've been here 15+ years and have never formally registered with a GP. We had private insurance for the first couple of years but, being fundamentally healthy, we stopped that after a couple of years. After that, for minor ailments we either went to the local pharmacy or the nearest GP. In some areas there are also local health centres which will provide emergency care or treat minor ailments.
Some differences between the UK NHS and here take a little getting used to.
With most doctors here, including specialists, it's OK to just turn up, wait and you'll be seen. Depending on time spent in consultation,as a private patient, you'll generally be charged anything from €10 to about €40 by a GP and €40+ by a specialist. As a State system patient you may be charged between €0 and €?????
As a private patient you pay the full cost of any prescription but if qualified for the State system only a proportion (I think 25%).
If you need blood tests, X-rays or scans etc., the GP or specialist will usually send you to a private biological lab or clinic (of which there are many). On completion of the tests you collect the results and take them back to the doctor for discussion. Results of simple tests are often available on the same, or following, day: others may need 2-4 days. You are responsible for keeping your own records, X-ray films, scan results etc. Some clinics are 'walk-in'; at others you need an appointment but the doctor will know.
Recent first hand experience: Needed lower back and hip X-rays. Without prior arrangement went to (allegedly) the most expensive clinic in the area. Reception advised there would be a 10-15 mins wait. Joined queue. Technician called me in (and recognised me from a visit three years earlier). Two X-rays taken; short wait then handed two X-ray films. I was on my way within 40 minutes, wallet lighter by €17.
Also 5 or 6 visits to GP with said back problem. First visit €40, subsequent visits nothing.
As part of annual service GP wanted ECG and ultrasound heart scan. Phoned recommended (highly qualified) cardiologist and made appointment for next day (remembered to take chest X-ray films). ECG, very thorough u/s scan, report of findings, brief lifestyle consultation took 70 mins. Left with ECG print-out, CD of u/s scan and written report for GP. Cost: €50 (with full tax receipt). 40 mins follow-up consultation 10 days later: no charge.