What freedoms did colonists obtain from the revolutionary war?

They obtained the greatest freedom of all: the freedom to run their own affairs. From now on, Americans would make their own laws, raise their own taxes, and determine the size and shape of their government. None of this was possible so long as America remained nothing more than a remote British colony.
One of the rallying calls of the Revolution was "No taxation without representation!" Most Americans hated the fact that they were subject to an often oppressive regime of taxation without having any political representation in the British Parliament. The British could impose whatever taxes they liked, and there was nothing that the American colonists could do about it. But all that changed in the aftermath of the Revolutionary War. Now that America was a free, independent country, she could ensure that those charged with imposing taxes were representative of the people themselves, and that if the people came to feel in due course that such taxes were too high, they could always reduce them, or even abolish them altogether.