Age Appropriate Infant Games
I know many people who consider newborns to be not quite human--kind of like blobs with unfocused eyes. Even those who don't see newborns as subhuman can find it difficult to develop a bond with them. After all, they can't converse, so how can someone find out what lurks inside that soft spot (fontanel)? I am the mother of 12 kids, and I suppose, unusual in that I favor the newborn stage over all others. As such, I've developed a kind of routine of what I think of as educational interaction. Here's a run down on how to play with your baby from birth through the first 9 months.
The best education she can have
Until about the age of 2 months, your baby won't show any interest in toys. All she wants is to watch you and listen to your voice. Keep your voice slow, steady, calm, and soft, and make sure she can see your eyes. When she breaks off eye contact, let her have a break of 5 minutes and then start all over again. Singing, cooing, and talking to your baby is the best education she can have at this point.
Keep toys soft
At around 2 months, your baby will discover her hands. As she waves them around, they catch her eye. If you put a soft rattle in her hand--she might hit her own head, so keep toys soft--she will start to make the connections in her mind: noise+hands, hands+movement, moving+me, my hands+moving+noise!
A delightful activity
At about 10 weeks, your baby will appreciate taking a swat at toys that hang above her. You can hang these above her crib, or dangle a toy above her as she lies on a blanket. This will be deemed a delightful activity for the next 4 weeks or so and will help her continue to develop the idea that her hands cause toys to swing and/or make noise.
By the time your baby is 3 months old, she will be receptive when you read simple books to her. Some books are made of interesting materials that your baby can mouth and touch. By the time your baby is 6 months old, everything she finds goes into her mouth. Make sure she has an ample selection of toys that are too big to swallow, but that she can experience with her mouth. Much of her learning at this age is done by mouth.
From 6-9 months, cups and water play are big winners. She can fill her floatable cups with water, dump them back into the tub, and fill them up again. This is a fascinating process and you might find that your baby doesn't want to leave the tub!