We registered our daughter a couple of weeks ago at RENIEC in Pucallpa. All we (father, mother and baby) had to do was to go there, present the baby's certificate from the hospital that she was born alive, my US passport (I could have presented my CE, too, if I had wanted to) and my husband's DNI. (Actually my husband's DNI had expired and the lady just typed in his DNI number into the computer.) The certificate from the hospital is only valid for 30 days. After it expires you have to pay and request another one from the hospital, so we were told.
As for the DNI, we were told that we had to make a payment of S/.16 (I think) at Banco de la Nacion or BCP, take our baby's birth certificate (which we were given when we registered her), and two pictures passport sized. We'll probably do the whole DNI thing when our baby can sit up on her own.
On a side note, I don't know how it is at other RENIEC offices, but the one in Pucallpa is such a HUGE headache. The lines are super long and even with the baby we had to wait a couple of hours. The lady who registered our baby told us that the system is really slow, which probably adds to the long delays. My husband went at 4am to go pick up his DNI and there were about 15 people before him.