Google Free in Manhattan/Central Park or other parks. Look up Time Out or NYU for free events. If you have a birthday, more freebies: google.
If you eat fruit, street carts are cheap. Many farmers' markets are in most localites
Avoid meal time rush in any restaurant, go earlier or later and look up offers.
Nice salad bars: Hale and Hearty. Substantial with cheese dressing, roll and no add-ons. Carry your own bag of nuts and seeds!
Food/wine demos at Williams Sonoma, Trader Joe's or elsewhere can provide free gourmet snacks. Check their events list. Apparently some chains, like Chipotle, clear out food on the last orders, i.e. serve huge helpings!
Home stays are useful IF you are flexible about other people's standards!
Re museums, many are free, some are free some days and some suggest donations: $1 is fine as they are subsidized by taxes, donors, etc.
Nat History: Meals are expensive in most museums but often the staff can recommend a nice, inexpensive place nearby.
Many have free days/nights. Do u have a press pass (if you are a press writer)? Do you have a Chase, Citi or other bank account? Can lead to free entry almost everywhere.
Many theaters have same day cheap tickets.
T Square food places/shops may cheat, have given wrong/counterfeit currency as change! Or charged more than the required tax.
Get a Subway pass: works on both bus/train. Check for student discounts and the papers/time required. For some discounts, you should ask months in advance! I give mine away after my last use if it is still useful for a few more hours beyond that time.
Senior/student/child discounts everywhere.
Have a super time in that amazing wonderful, inspiring city!