We have just signed up for the Clinic in Monte Grande for our children (I live just past Monte Grande, so it may be far for someone living in the city, but I know there are several similar types, and I think the Clinic we have does have an office downtown). For the both of them it is like $103 (pesos) per month, and there are small copays of like $5 pesos a visit with our specific plan. It includes dental care as well, and it's a bit like Kaiser in the states in that one place does it all, and I'm to hope that the difference is they take better care of their paitents than Kaiser! They will make outside referrals if necessary, and work with area hospitals. My father in law had a stent put in at the clinic, and spoke very highly of it. They also will send you a doctor to your home (and a pediatrician in case of a child), and work with an emergency ambulance service. The woman who enrolled us also came to the house. The kids do not yet have DNI's, so that wasn't an issue.
We took our daughter to the public hospital about a week after we arrived because she slipped on the kitchen floor, and landed flat on her face. The hospital did x-rays, and we were seen faster than in an American hospital, but they didn't seem to be too busy that day in particular. They never asked us more than her name, and their was no paperwork.
IMHO, I would wait until you settle in, and know that you can use the poor run down hospitals in case of emergency. Also, the pharamacist is a psuedo doctor for minor ailments. Everything is under prescription supposedly, but if you ask, and it isn't anything too strong, they'll give it to you.