When will my baby's hair start to grow?

It differs.... Every baby is different. Some will be born with a full head of hair and keep it, some will lose it and regrow it early and some wont grow significant length of hair until 2 years old. It just really depends on the individual baby...
Sooner or later. Sooner or later, a baby will have a nice head of hair. The color, amount, and location of hair is usually genetically determined, and thus is a familial trait. Some babies have lots of hair at birth, while others have very little hair. Some don't seem to lose their newborn hair, while others do (and go through many months of little hair). At check-ups, the doctor can tell if the hair looks ok.