When I moved to NYC, I was really excited to check out all the great performances in the city. Unfortunately, a lot of those cultural experiences are pretty expensive. This has not deterred me, however, from going to the Metropolitan Opera and to Broadway. Both have ways to get cheap tickets if you nose around enough. (Although admittedly, cheap is a relative term here)
The Met has a few different options for acquiring cheap seats. Thus far, I've only sat in seats with a partially obstructed view. You miss a little bit of the action, but the family circle box seats have excellent acoustics. When I saw Carmen, an old man sitting near us said he always got those seats because the sound is the best in the house. Another option is purchasing family circle seats (the highest level in the opera house) way ahead of time - I already have tickets for a February performance that cost the same as the box seats I sat in last month. Those were $30 each, plus some annoying ticket fees that jacked it up to $40. Other shows have less expensive tickets though - Turandot, for example, has $20 box seats.
Two other Met options that I haven't taken advantage of yet are Met student discounts, where they release a certain number of tickets that can only be purchased if you have a valid student ID, and standing room only. A friend of mine has done standing room only, and says it's not that bad, but I don't know whether I can handle standing through a 3 hour opera. I'm a student member at the Met, so I get emails about tickets, but haven't bought any yet.
Traditionally, to get cheap Broadway tickets you need to do rush/student rush. I have sat in student rush seats, but a friend went to the box office for both of us so I have no experience with the lines. We sat in partially obstructed seats for How to Succeed in Business Without Really Trying last summer, for $20 each I think, and had a great time.
My sister visited this past weekend, and we discovered an option at least for shows at Lincoln Center Theater: LincTix. You have to be between 21-35, and you can sign up to get $30 tickets for almost any show, for any seat still available in the theater (as far as I can tell). We bought tickets to War Horse on Friday night, and sat in the front row of the Loge (the balcony) for the Saturday matinee. Those are usually $125 each, but we payed just over one quarter of that amount. And the show was very good.
City living definitely isn't cheap. But thus far I've had pretty good luck finding discounts; all it takes is a little bit of extra work.