According to Goat World, a baby goat is called a kid. A kid is considered any baby goat under 6 months of age. The term is used for both male and female goats.
Female goats over the age of 6 months are generally called doelings, and are female goats that have not reached maturity. Male goats over the age of 6 months are called bucklings, which are simply immature male goats. Once a female goat has reached maturity, she is called a doe. A mature male goat is called a buck. When a male goat is castrated, he is called a wether.