Immerse yourself in the genre. Even a child can start on the road to Broadway by living and breathing theater. Attend plays--musical and otherwise--local, regional and beyond. Know what's hot today and what gained rave reviews yesterday. Watch the actors and watch their careers. See what they do well and not so well.
Act. Before you make it on Broadway, you will need to make it in a hundred other venues and shows, playing a variety of roles in a mix of plays and performances. Any chance you can, audition for shows. Participate at every level. Understand not only what the role of actor is, but also the workings of the theater; know who does what, from the directors to the technicians.
Study the craft. Learn to sing and dance. Take voice, dance and acting lessons. Take drama and public speaking courses in high school and join the drama club. Perform in productions in and out of school. Consider your college choice and courses with an eye to your future career. Choosing a university near New York City with an emphasis on the performaning arts would be a good place to start (see Resources below).
Work outside the craft. However you choose to pay your bills while you await your shot on stage, try to interact with the public as much as possible. Particularly while working in the Big Apple, you never know who you might be serving across the counter. Contacts are the lifeblood of theater advancement, and you ignore that fact at your peril. Check out the website in the Resources section below for theater jobs and contacts.
Promote yourself. Get a real resume and professional headshots together and be ready to introduce yourself and your talents to anyone willing to listen. Get an agent and go on every audition she finds for you.
Audition, audition, audition. The only chance you have to make it to Broadway is by trying, over and over again, to get the part. Come fully prepared, in body, mind and attire, for every audition, but be prepared also for lots of rejection. If you cannot handle rejection, you will never make it to Broadway.
Learn another theater skill. While you await your big break, working in set design or makeup can at least allow you to be in the right place when the right time comes your way. And you cannot find a better place to make theater contacts than in the theater.