Check story descriptions for all current Broadway shows at sites like broadway.com or broadwayworld.com. At Broadway.com, click on "Find a Show" at the top of the page. Links are available to individual shows, as well as links that divide the show themes into genres so you can tell which ones are kid friendly or headlined by celebrities. This is a one-stop way to determine what kind of production appeals most to you.
Click on the "Buy Tickets" link for the show that interest you at any of these sites. This does not commit you to a purchase, but does show the range of ticket available. Most of these sites also have a link for ticket deals. Broadway.com has one called "Last Minute Deals." In August of 2011, it offered 23 percent discounts on next-day performances.
Read reviews if you're concerned about the quality of the show. While reviews can be a great guide, they should not be the only factor in your judgment since some shows take a little time to get things right. Most major newspapers in the New York area have archives of reviews. Some individual reviewers also have sites. One of them, Ben Brantley, also has links to the individual shows with more reviews by other critics. Brantley's site, Did He Like It.com, also has links to ticket sellers.