Most Americans in the summer of 1776 learned the news of independence through newspapers, as well as broadsides and public readings. The first report of the Continental Congress declaring independence was published in The Pennsylvania Evening Post on July 2nd, followed by the text of the Declaration in the July 6th issue. Philadelphians would have learned about independence almost immediately, while New Yorkers found out three or four days later, Bostonians almost two weeks later, and South Carolinians almost a full month later.
Check out this animation of how the Declaration of Independence spread through the newspapers!