Read newspapers regularly. Go to the library and read about political organizing. If you are interested in running for the school board or city council, attend their meetings. You will see the process first-hand and make valuable contacts for your own campaign. Choose the office for which you wish to run.
Many people start with a small, local race.
Find out which officials are planning to retire or run for another office, leaving their seats open. It is almost always easier to win an election for an open seat. Research the problems and issues previously faced by the holder of the office you wish to hold. Before you begin your campaign, understand campaign laws and finance rules. You will need to contact the county clerk, the circuit clerk, the Secretary of State's office, or the Federal Election Commission, depending on the office you seek.