There are no educational requirements to be a politician. All you need is support to run. Who this support comes from depends on the office. Once you have enough people that support you, they need to sign a petition to add you to the ballot. You pay a fee and, presto, people can vote for you.
You, of course, need to campaign.
Once you take office, if it's a paying office (some local positions don't pay), you'll get paid almost immediately. Even where it pays, most local offices don't pay very much. Unless you make it to Congress/Senate or are governor, don't expect much money. Even so, it's not a lot when compared to what's possible in the private sector.
You'd better your chances, of course, by studying law. This is a credential that might help you, but it's definitely not required. Also consider political science or history.