If you want see a lot and have some flexibility, you should join a club. My favorite is Play-by-Play.This was a genius idea by founder, Hugh Hayes where he teamed up with hundreds of venues and organizations to fill their seat, and he his able to offer them to the public for a small service fee. For $99 a year, you have a lineup of broadway, off-broadway, off-off broadway shows, cabaret, and more, and most for just a small service fee of $4.50. This is how I got to see so many Broadway, Off-Broadway and Cabaret shows for virtually free. No lie.
You can easily makeup your membership fee seeing only one show.
There are a lot of ways to get discount tickets for NYC theatre and entertainment. Here is some guidance
There is also Theatermania Gold Club. It's the same cost as Play-by-Play and they sometime have different shows. You can join both (again, this is well worth it for the avid theatre fan), but if you don't have the budget, Play-by-Play would be my club of choice
You might qualify for Theatre Development Fund. They do offer many mainstream shows, but the tickets cost a little more that the previously mentioned clubs, but you can get a Broadway ticket for select shows for a little as $41. It's only $34 to join, but you must "Qualify". Visit the TDF Website They also run the TKTS line in Times Square, and offer same day tickets for up to 50% off. Take note that there is a separate line for "plays", and its usually much shorter than the "musical" line, so get to know your lines. There are representative out there to help you in their signature RED color, so don't be afraid to ask questions. For more information on TKTS click here
There are many discounts available without membership. Many shows are discounted up to 30% or more. And sorry, you are not going to see Book of Mormon on these discounts lists
You should subscribe to one of more of these websites so that you get early notification of ticket sales, special sales and discounts. You can usually get an orchestra, mid-week ticket while a show is in previews for under $100.
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The best times for discount tickets are mid-week (Tues-Thurs).
Some of the more popular shows have lottery, rush, and student rush and Standing Room policies but keep mind these are not guaranteed, so have a back-up plan. These are highly discounted. I like this playbill article that gives a pretty good overview of who, what, and how much.