CHOOSING a show:
My current Broadway favorites
I keep a list of some of my favorite recommendations on the Stage Directions page (but please do some further research before you decide! :)). You can also find the kid-friendly suggestions on the Stage Sprites page. You'll also find current theater news under Upstage...Theater News Feeds.
Check out Playbill.com
A comprehensive website for finding a show is Playbill.com. They have a listing of every Broadway show that has been announced (including those that are scheduled but not selling tickets yet, and even ones that do not have a theater yet; as well as Off Broadway shows). They also have a weekly schedule of shows with days/times~invaluable when planning. One of my favorite Playbill resources for planning ahead is the schedule/list of upcoming Broadway shows.
Visit the show websites
Where possible, I've provided links to the various show websites on this site. The show's site is a one-stop shop for the practical (schedules and ticket links), and the best way to get a feel for the show, with videos, audio, synopsis, cast bios, etc.
Periodically check out the theater news websites I've listed on this page for the updates about shows in the works, casting and special offers.
Trust your own interests/sources
Keep your eyes and ears open. You might hear about your favorite actor planning to appear on Broadway from watching TV or reading the newspaper. CBS Sunday Morning, (in my opinion, the best show on television), routinely does stories about current and upcoming shows on Broadway.
Weigh the discount options
Also, before you look for discounts, get an idea of the shows that appeal to you and then target your search. You may find that it's worth more to pay the extra money to see a show you really want to see, rather than compromising on a discounted ticket to a show that isn't up your alley. That said, discount tickets are a fabulous way of experiencing shows and theater that you might otherwise not expose yourself to because of the expense of the full price ticket options.
The invaluable site theatreaccess.nyc offers extensive information about finding shows that will accommodate various limitations, including mobility, audio needs, autism-friendly, etc.