Health service registration
After NIE and residency you can get the health card
You need to register with the local office de Sanidad which then gets you a health card to say you are permitted to use the health system - you need an NIE and the proof of residency in the district for this. Up until this point you are probably covered by the E101 from the UK, but I wouldn't rely on it.
Once you have the health cards you can go to the local CAP which is the equivalent of a primary healthcare trust in the UK where you can register with the doctors.
We did find one quirk. My wife had her NIE first, so when we got the health card the children could be put on her card, but I couldn't without having my own NIE. As I got my NIE sorted out later and am still lacking a few docs on this front.
Once you are registered you can use the health services much as you would do in the UK