Here are 6 positions that are a good starting point for your first job in politics. While they're not certainly not glamourous, and will typically constitute long hours with meager to no pay, these jobs are essential to politics as a whole and the foundation of your successful career in politics.
Campaign Intern Call the state party and find out who the most competitive seats in your area are, and then call that campaign and say you want to help out today. Go down to the office and start volunteering, most likely by licking envelopes and making phone calls.
Government Intern Apply for an internship in a government office, often unpaid but sometimes not. Even a two-month internship can demonstrate that you are a reliable person whom they like and enjoy having around. That can lead them to offer you an unannounced position or a position shortly opening, and help steer you to it by telling the hiring manager to pick you.
Journalism Intern You must have a portfolio of some kind to get started, so start writing for a local paper or a somewhat prominent blog. Keep a clipping of your best work, and try to write original content that you do actual reporting for. Build up a file of 6-12 really solid pieces that you're proud of, and then find someone who can be critical and give you several ways to improve your portfolio. Call up the section editor for a paper you want to write for, and tell them you'll write something "on spec" for "no kill fee" meaning that there's no penalty for them if they decide not to run the article. Then, do a great job on those initial articles and become a regular with that editor. Once you can get 'in' there, and demonstrate your ability to write, ask your editor to keep an ear out for any openings. In time, they'll find a place for you if your writing is good.
Legislative Office Intern Definitely the best bet is to start here as the intern and work your way up. Some legislative staffers think interns are a waste of their time, and so they may not care enough to actually call you back. This happened to me once so I called every day for six weeks about an internship, which they eventually gave to me as a college freshman, likely just so I wouldn't call anymore. Many staffers, though, like having interns, even if only to make themselves feel more important. And truth be told, there are plenty of political science majors who take these positions and sneer at them, or treat them lightly, but demonstrating a hard work ethic and a positive attitude will go far. One thing that often gets overlooked are tours in the state capital or the national capitol. And this is a huge staff mistake. Tours are a constituent's primary interaction with an office, and each one of those people knows who your candidate is and votes, which puts them in the most precious box of electoral calculus: the reliable voter. So make the tour great, read up on the history of the building you give tours on, ask for permission to lead the constituent tours, the staff will think you're crazy, but you're crazy like a fox. If you give great tours and excel at this job, which I can confidently say you will based on how lazy most staffers are about it, you will make a great impression on management. And if you don't give tours, or that doesn't work out, find one project or item that you can really excel at. I got a job offer once because I was amazing at constituent mail, and had to be told by the Chief of Staff in a Senate office that my letters were getting noticed too much, and that as an intern I was way overstepping the bounds by asking for casework. That's the kind of "problem" that leads to a job offer, where you're doing too much work and take your roles very seriously.
Major Donor Staff These jobs are extremely hard to come by, you want to get to know a major donor and impress them to the point where they want to keep you on staff for them. The difficulty is that these people can also command people who have a longer history in politics than you to take these jobs because they're seen as such a safe bet. So, have serious skills that are well-demonstrated, and find a major donor to start a relationship and friendship with where they know and understand your skills. When that happens, in time, they'll create a job for you, but you can't ask them for it or else they'll see you as desiring a handout, which they won't want to give you.
Non-Profit Intern Almost every non-profit is short-handed, so go in and say that you want to help out. Do your best, put your heart into even the trivial details, and be continually value-added and they'll see and appreciate your value. Don't let them continue to use you for free help, however, and demonstrate that you want paid work to continue. They'll either find the money to pay you or they won't, but most likely they will if they see you as value-added, especially if you have fundraising skills.