If you're interested in moving into a position with management responsibilities, the best thing you can do is to start seeking out leadership opportunities in your current job, even though it will likely mean more work without greater rewards—for now. Stepping up and taking on additional responsibilities will give you experience that will pay off later.
For instance, you could volunteer to manage interns, lead a team project, train new employees, or screen job candidates. Many employers will be grateful that you're volunteering for these jobs, and even more grateful if you show that you're actually excited to do it.
The payoff for you? You'll be expanding your skills, proving yourself to your employer and colleagues, and establishing a track record of doing well in this area ... which will pay off for you down the road when someone has a higher-level opening and remembers being impressed by you. (That assumes that you do a good job, so do a good job.)
Additionally, make your desire for this kind of experience explicit to your manager. Good managers will try to help you identify opportunities to gain experience in these areas.
Take initiative, be open about your goals, volunteer, and work hard. You'll likely end up with the experience you need.
Alison Green writes the popular Ask a Manager blog, where she dispenses advice on career, job search, and management issues. She's also the author of Managing to Change the World: The Nonprofit Leader's Guide to Getting Results and former chief of staff of a successful nonprofit organization, where she oversaw day-to-day staff management, hiring, firing, and employee development. She now teaches other managers how to manage for results.