Legally speaking, we do not have any real responsibilities as citizens of the United States. We are required, of course, to obey the laws. Beyond that, however, there is nothing that we are positively bound to do (as opposed to things that we are negatively bound not to do).
Morally or ethically speaking, we do have responsibilities as citizens of the nation. Since the nation protects our freedom, we have a responsibility to help make the nation a better place to the best of our ability. That would include doing things like volunteer work or charity work. It would also include things like voting and making our voices heard in other ways so as to try to bring about change in our nation.
Overall, then, we do not have much in the way of legal responsibilites, but we do have a moral or ethical obligation to do our best to improve the country.