Hatsumiyamairi (a Shinto rite of passage for newborns) (初宮参り)
Hatsumiyamairi (a Shinto rite of passage for newborns) is to thankfully report to Ubusunagami (guardian deity of one's birthplace) that a baby safely had the one month anniversary. In general, miyamairi ((the custom of) taking one's baby to a shrine (to pray for blessing)) refers to Hatsumiyamairi.
Summary
In Japan, there is a custom for parents of a baby to take their baby to a shrine, celebrating the baby's birth and wishing the baby's healthy growth, with a grandmother on the baby's father's side after one month from the baby's birth. Nowadays, it is common that a grandmother on the baby's mother's side accompanies them to a shrine as well as the grandmother on the baby's father's side.
In Kyoto, there is a custom for a baby-girl to go to a shrine earlier than a baby-boy, wishing that a baby-girl would marry early.
Dressing
The formal dress for a baby is wearing a congratulatory kimono jacket with a family crest on top of white 'habutae,' a thin, soft, durable Japanese silk Kimono. It's desirable that the baby's parents are formally clothed as well.