Personally, I have only had GP appointments in the state health system here in Andalucia, but know quite a few people who have been treated in the state system for serious illnesses such as cancer, heart problems, etc. and they have all been happy with the service. I have visited several people who have been hospitalised in the state system and been impressed by the cleanliness of the hospitals and the fact that patients are accommodated in rooms with no more than 2 beds and a bathroom shared between 2 patients.
I find it very easy to make medical appointments at my local health centre, in Andalucia there is a central system which covers every health centre where you can either ring one central number, or make the appointment online (which is what I do), or by text message. Same day appointments are normally available, and doctors will make home visits if requested by 10am if a patient is too ill to get to the health centre. We have two health centres in our town, one is open 24 hrs, 365 days a year (they have an Emergency Department which is always staffed) and the other is open from 8.00 am until 9pm. Doctors do work within time constraints for appointment slots, though, just like in the UK. There is little time for the social niceties or general chit chat, the staff are very professional and businesslike and you just say why you are there, answer any questions the doctor asks, they issue a prescription or request for tests and that's it, next please.
Do not underestimate the language problem, if you don't speak Spanish. Translators are not provided on request, at no charge, as they are in the UK. If you can't communicate in Spanish you are expected to arrange and bring along your own translator (although in Andalucia there is a telephone translation service, but I have never heard anyone say they have used it). In many hospitals (but not all, and not in health centres) they have a team of volunteer translators who can help in various languages.