I'm the author of Experimenting With Babies: 50 Amazing Science Projects You Can Perform on Your Kid, which I hope you'll check out. The book takes published academic studies in various fields of child development -- primitive reflexes, motor skills, cognitive development, language development, social/emotional development, and more -- and adapts them so parents can perform them at home on their own baby, with no special equipment needed. You'll learn how to perform some famous time-tested experiments, as well as new experiments that are adapted from very recent research.
Below is an example of the type of experiment you'll find in the book. It's based on a November 2011 study that deals with perception and cognitive development.
Perform this experiment when your baby is between 13 and 15 months old.
For this experiment, you'll need two adults: one to participate in the experiment and one to observe it.
Have one adult sit at a table on which two similar-looking toys are placed, one to his left and one to his right, and place your baby in a high chair opposite him.
Now, you'll perform four 15-second sequences, each separated by a pause of about 15 seconds. For each sequence, the observer should keep track of which of the two toys your baby looks at longer.
1) The adult should turn his head and gaze toward one of the toys for about 15 seconds.
2) The adult should repeat the action, gazing toward the same toy for another 15 seconds.
3) The adult should place his hand between the two toys and stare at the hand for 15 seconds.
4) The adult should exit the room, leaving the two toys undisturbed on the table.
During the first and second sequences, your baby will look longer at the toy the adult gazes toward.
During the third sequence, she will continue to look longer at the toy the adult gazed toward.
During the fourth sequence, she will look slightly longer at the other toy.
In a 2011 study, 14-month-old babies were shown video clips in which an adult gazed toward one of two toys.
Then, one group of the babies was shown a video clip in which the adult stared at her hand, rather than either of the toys, while another group was shown a video clip in the toys were present but the adult was not.
The study found that the babies in the first group looked much longer at the toy the adult had previously gazed toward, but the babies in the second group looked slightly longer at the other toy.
The study's author says the results support the idea that 14-month-olds form an association between a person and an object after having seen the person gazing at the object, because the babies preferentially looked at the object even after the person had stopped gazing at it -- but only while the adult was still in close proximity to the object.
Paulus, Markus. "How infants relate looker and object: evidence for a perceptual learning account of gaze following in infancy," Developmental Science 14:6:1301-1310, November 2011.