Kids can be found in Stardew Valley from the start, but having kids of your own is also possible. In order to have kids you first need to have a wife or a husband. They also need to be happy, meaning they have at least 12 hearts. To build a nursery you need to visit Robin's Shop and to get the second upgrade (that adds the nursery), you will need 50,000 gold and 150 hardwood.
If everything above is done a cutscene will occur when you go to bed (thought there is a small chance of getting the scene) where your partner will ask if you want children. Saying yes will result in you or your partner getting pregnant or your partner filing papers for adoption (for same-sex couples). After 14 days, your partner or you will give birth and an infant will appear in the nursery. After an additional 14 days, the baby will grow into a toddler. It is possible to have 2 children. The couple will have one daughter and one son. Once the child is born, the sex will be revealed and then the player can name the child. You cannot change the name after you have pressed OK.
Time doesn't seem to be a factor. The kids cannot get any older other than their toddler stage.
