People can become good citizens by practicing justice and fair play, helping others and observing rules and regulations.
More than just citizenship
Whether you're born with citizenship or earn it after immigration, there's more to being a Good Citizen than just having that legal piece of paper. It's about how you conduct yourself as a person, how you interact with your community and society, and how you pass what you know and learn on to others. This isn't a discussion of patriotic brainwashing or anything along those lines; this is a discussion of the education, interpersonal skills and common courtesies needed to keep the country moving smoothly and strongly.
\Starting with education
People who know more understand more. You don't have to be a PhD student in an Ivy League University, but you have to cultivate a habit of learning-- ignorance leads to fear, and fear leads to easy manipulation. A Good Citizen isn't easily manipulated, but understands what's going on and can make rational decisions.
A few useful topics to keep up on: Politics, both local and national; Education; the Environment; Energy Issues and what can be done at home about them; How things like banks and investments work; the actualities of Tax and tax Reform, not just the hype; How the government works and the best ways to vote it to work better; Heathcare; All the ways a ruling body helps it's citizens-- schools, hospitals, community groups, local clean-ups, the Postal Service, and so on.
\Sometimes these things are boring, but understanding how they work will keep you from being snowballed over later.
Meanwhile, find a few topics you really care about on any subject, and keep learning. Research their histories, follow their trends, know their inner workings and become active in their communities: Know your passions and all the facts, and you can defend them when other things start infringing. Know how things worked and went in the past, and you can see them repeating in the present. All learning expands your view of the world and how it works, so keep learning always, and keep up with a rapidly changing world.
Social and Community Responsibility
A Good Citizen has to think about the nation as a whole as well as their local community and their own families. Americans tend to be sort of self-centered and short-term thinkers, but if more people thought about how their actions and their decisions affected the rest of the nation, we'd have better planning and more reasonable long-term goals, instead of being reactionary and scrambling to fix unforeseen damage made by short-term planning.
Aside from being informed and educated, and helping your kids to be the same, here are a few ideas for societal thinking: