Why go to New York City for a Broadway show? Production values are the highest, you might see the "original" cast, and you might catch a star.
Before going to a Broadway show, you should ask yourself some questions: Do I want to see a new show? Does it have to be straight theater or must it be a musical, or will I be flexible? What is my budget? Do I want to see a famous star?
Then read "Time Out New York" online or "New York Magazine."
If you want to see a star, here are some shows with big names from TV and movies: Al Pacino is in David Mamet's CHINA DOLL Both Bruce Willis and Laurie Metcalf (known for "Roseanne" and "The Big Bang Theory" on TV) are in the stage version of Stephen King's MISERY. Kelsey Grammer is in FINDING NEVERLAND while Tim Pigott-Smith (known to PBS viewers for "The Jewel in the Crown" seen Saturday nights on WNED-TV) is in KING CHARLES III.
You can always go to 47th Street and Broadway for tickets where they will be discounted 50% (but note that's 50% off the top ticket price). If you go to the individual theater's box office, here are some "tricks, tips, and traps" to help you. You can almost always get a single ticket. If you are a couple, ask for two singles. Let the theater know that they can be separate. You might not sit near each other, but you'll get in. Ask if there is standing room. Almost all theaters have standing room at the back. Ask if there are obstructed view tickets. Even the worst seat in the house might yield a great night of theater. And look for people with clip boards. They know about cancellations and such. They will start quoting ticket prices for their theater. As it gets closer to curtain, the price will start to drop. When it hits your budget, make your move. BUT DO NOT BUY FROM SCALPERS. Buy from the box office.