Exactly what happens depends on the denomination and the parish. For my daughters (Catholic) it was mass with the children dressed appropriately (white dresses and sometimes veils for the girls, suits or school uniform for the boys). The children, with their parents, were in reserved seating at the front of the church, and when the time came went up first. Friends and guests were at the back of the church.
Probably the most suitable gift is a missal, but check with the child's parents - they may have already bought one and may be able to suggest something else.