Every state in the United States has state-owned parks, monuments, historical sites or other places of interest that might benefit from a clean-up effort. Cleaning up state highways or bike and hiking trails is another idea. Contact the state office in charge of the location where you want to clean, and ask about how you can set up a volunteer effort to beautify the area.
Donate your time to give aide to those who need it most. State projects, such as state-run nursing homes, animal shelters and hospitals, often have programs in place to facilitate a group that wants to help. Your group or class can perform maintenance on buildings and spend time with people in a nursing home or animals in a shelter.
Once your group decides on a cause, chances are high there is a state program in place to support or fight it. A letter writing campaign to a state official is one idea. Perhaps this could be in response to a state decision that impacts your group, such as a cut in funding to schools or the closing of a state-run program. Peaceful protesting demonstrations are also within your civic rights.
Voting on the state level is just as important as it is on the federal level. Your group may like to research the candidates in a state-wide election and throw support behind the one that best supports your interests. Another option is to volunteer to help citizens vote, regardless of political affiliation. Volunteers are often needed to man the polls, make phone calls and drive people to vote.