An interest in politics is a very broad statement. You can get involved in helping/running campaigns, work at the local elections board, or be a candidate your self. Every where has loca political parties. Each local party typically has a monthly meeting. You can attend the meeting as a visitor to see how politics work. Once you are able you can get elected to a position or volunteer with that local political party. This gives you experience in the candidate process and let's you easily meet other local elected officials. Typically local campaigns also have volunteer opportunities. Stuffing envelopes, making phone calls, etc. volunteering for a campaign gives you political experience from the campaign side of politics.
Typically those who want to work for campaigns or county election offices have a political science degree. If you are interested in being a candidate or elected official, there are no specific degree requirements. However, having local political experience will be one of the most valuable things that you can do.