This question is so complex to answer that the UK and other governments have introduced classes into the school curriculum to help children learn about the rights and duties of citizens. The laws on citizenship vary from one country and one era to another, as do popular perceptions of what makes a "good citizen."
Even so, there are some constant factors. Most countries actually have a law stating that it's illegal to witness a crime and do nothing at all to prevent or report it. This points to an almost universal underlying principle summed up by John Donne over 300 years ago: "No man is an island." The basis of citizenship is this recognition that we cannot act as if we were alone on the planet; that our freedoms end where another person's freedom begins; and that if a fellow-human is in trouble, it's ultimately in our own interest to help them.
The difficulties lie in deciding what, if any, help to offer, keeping a balance between freedoms and duties, and dealing with people who persistently demand rights without accepting responsibilities; these are some of the challenges facing a responsible citizen in a democratic society.