Congratulations !!
You should be covered by your partner’s social security. No doubt there are small differences in procedures depending on your autonomous region but if you have an NIE number and are registered with your local council at the same address as your partner you can get a health card which is covered by his payments. I put my partner on my card when she stopped working, we needed her DNI, my NIE, my social security card and the “volante de empadronamiento” from our council. Our local health centre (ambulatorio) had the form we needed to fill in and once we had all the paperwork we haded it back to them. It took a further couple of weeks for her to receive her health service card but was still covered.
If you are an EU citizen with a NIE and registered with a council you will automatically get emergency healthcare within your autonomous region, I beleve that this also covers pre-natal and childberth services, but best is to get yourself on your partner’s card.
You should do this asap to get youself signed up to a midwife, from the service we received from Sergas in Galicia, we can recommend childberth on the public system.