The rights and duties of a citizen are to be a good citizen and abide the laws in the Constitution. You have to support and defend the Constitution, serve the country when required, participate in the democratic process, respect and obey federal, state, and local laws, respect the rights, beliefs, and opinions of others, and participate in your local community. The rights that you have are you can vote in federal elections, serve on a jury, bring family members to the United States, obtain citizenship for children born abroad, travel with a U.S. passport, run for federal office, keep and bear arms, and become eligible for federal grants and scholarships.
Being a citizen of the US comes with a lot of great responsibilities. A few are Paying attention to what is happening in the world, Voting, paying taxes, speaking up for what you believe, education, volunteering, have faith in what the people have chosen, obey the law, adapting to new things or laws, jury duty, and for some being in the military is a big one. There are plenty more than that, but I would have to say that paying attention and voting are the two most important by far. Voters are who pick the people that run the country you live in. If voters don't know what is happening or who can fix it, their uneducated vote could change the future. It is all about the responsibilities of the citizens.
Obeying all the city, state, and national laws
Paying all your Taxes
Jury Duty (when needed)
Serve as a Witness on jury
Register for the Draft when you turn 18