It's a wonderful moment when your baby first reaches for you. Not only is it an obvious sign that she's physically capable of doing it, it also means she's come to recognize you as a person she needs and trusts. Reaching out for you is usually a sign she's ready to be picked up and cuddled.
Vision and Development
There are two reasons newborns don't reach for you right away. The first is they simply don't have the motor control to move with purpose. The second is babies are born with better peripheral vision than focus vision. When your baby can see you more clearly, you'll notice his interactions with you will increase.
Babies may begin reaching for objects as early as three months, according to Parenting.com. Early on, her reaching will be more general. Her world is full of fascinating objects, and as soon as she learns she can touch and examine them at will, she'll have her hands on everything. By six months she may begin holding her arms out to you in a conscious effort to be picked up, notes Boston Children's Hospital.
You've got something else to look forward to at this age: hugging. Once your baby begins interacting with the people in his little sphere and recognizing those who he feels happy and safe with, he'll begin giving hugs. Some babies are cuddlier than others, so don't be too upset if yours isn't quite ready to have a snuggle fest. Even if he isn't very touchy-feely, he'll probably be receptive to hugging before bedtime.
This is a great time to encourage playtime on the floor. Put her on her stomach for short periods of time so she can strengthen the muscles in her neck and arms. Then place toys just out of her reach, to prompt her to reach for them. Pretty soon you'll have a crawler on your hands.