If you're renting a place with a kitchenette, by all means, cook. Supermarket food is inexpensive, and there are a lot of outdoor markets and organic shops in the city. In summer and fall, head to the Greenmarket on 14th Street for farm-fresh fruits and vegetables, plus homemade honey, pastries, and jams.
If you're stuck with restaurant food, the key to cheap eating is to get out of the areas where tourists congregate. Even walking two blocks in any direction will make a big difference. Many restaurants post a menu outside, so you can check prices before making a decision.
Ethnic restaurants and buffets are also economical alternatives. Many diners (especially the ones in midtown) have been preserved intact since the '50's and are not only affordable but also an experience not to be missed.
In January and June, many restaurants participate in "restaurant week," offering fixed-price lunches for $20 or less.
Entertainment
Broadway
No visit to New York is complete without a trip to one of its many theaters. Tickets to premiere Broadway shows are hard to come by and usually too expensive for the budget traveler. Broadway shows that have been around for a couple of years make discount tickets available at various TKTS booths throughout the city. Same-day tickets are sold at a 25 to 50 percent discount. You can also buy standing room tickets (and you'll be nicely located directly behind the orchestra seating) a few hours before the performance, but only if the show is completely sold out.
Free (or almost)
For free performances, you can check Shakespeare in the Park, a first-rate performance at the outdoor Delacorte Theater. The atmosphere is wonderful and big-name stars are known to join the cast every summer. The Carnegie Hall offers partial-view seats to most of its performances for $10, and the Metropolitan Opera sells standing tickets for $20 a week in advance of a performance.
Museums
Many NYC museums have days where tickets are either free or heavily discounted. The Guggenheim, for example, offers a two-for-one ticket deal on Fridays after 6pm. Other museums have a suggested admission price (the American Museum of Natural History is an example, with a suggested entrance fee of $15), but you can usually pay as little as $1 and still get in.
Nightlife
Heading out at night? Women get into most nightclubs for free. In some clubs, entrance is always free before midnight, but after 12am, everybody pays. Happy Hour may run until 10pm or later in some bars, and you may end up paying less than $2 for a pint of beer.