From three months, most babies will begin to try to co-ordinate their bodies with the things they see by attempting to grasp objects in view. They will happily lie under a 'baby gym' or sit propped up in a baby chair or car seat, trying to swipe at toys in their field of vision.
You can help hand-eye co-ordination by holding a rattle about 10 inches away from your baby so they can focus on it. Move the toy slowly and watch them 'track' it with their eyes as they move their head simultaneously. Soon they will reach out a hand towards the toy.
Although by three months babies can open their hands, most three-month-olds can also lift their incredibly heavy heads right up while lying prone. Later they can lift their shoulders. As they do these little 'press-ups', they will start to scratch at surfaces with their fingers in an attempt to pick up patterns from the carpet - as yet they don't know the difference between pictures and 3D objects. By the time they reach six months, most babies have achieved sufficient head and neck control to sit up for a few seconds unsupported.