This may vary according to the Regional Authorities, but in general, after registering with the Social Security you get only a document that shows this registration.
You then need to apply for the "Tarjeta Sanitaria" which is a card that shows you are entitled to medical assistance by the Social Security system and registered with your Regional Community, by signing the proper application form. The Tarjeta Sanitaria will generally come in the mail a few weeks after filing the application form.
Check with your local SS Authorities since the procedure may vary in each one.
Google "Instituto Nacional de Seguridad Social" for more information in Spanish about the general Social Security system.