I lived in NYC for about 8 years - It's a little hard to make suggestions without really knowing you and your interests, but there's TONS of cheap things to do in the city.
Generally speaking, for food: If you're staying in Times Square, I suggest you walk over to 9th Avenue (anywhere between 42nd street and 54th street) for food. You'll find a huge variety of options (Italian, all kinds of Asian, diners, Southern, Greek, whatever), and they'll be cheaper than anything you'll find in Times Square (or east of Times Square) itself. A couple of favorites: Delta Grill (Southern), Westway Diner (it's just a diner, but good food and soooo cheap), Beans and Rice (Brazilian.) There's a good Chinese place in the same block as Westway (between 42nd/44th) - I don't remember the name.
If you're interested in theatre tickets at all, at the north end of Times Square is the discount ticket booth, TKTS. (http://www.tdf.org/TDF_ServicePage.aspx?id=56) You stand in line to get same-day tickets, up to 50% off. The shows available vary from day to day and hour to hour, but you can get a really great deal.
Museums:
Natural History is amazing, especially the Rose Planetarium.
Metropolitan Museum of Art, of course
Guggenheim - modern art
If you're really into seeing "sights", this is definitely a good deal:
http://www.citypass.com/city/ny.html
The Circle Line tour (boat tour around Manhattan) is fun and really interesting, a great way to get a sense of the whole city. It's a little spendy, but I think worth it.
http://www.circleline42.com/
Some suggestions of free things to do:
Walk across the Brooklyn Bridge. You can take the subway downtown, get off right at the Manhattan end of it, walk across, have lunch in Brooklyn Heights (very hip little neighborhood) and then hop back on the subway back to the city. Great, great views, great people-watching.
Shopping - go down to Wall Street to Century 21. http://www.c21stores.com/ It's a madhouse, but you can find some really fabulous deals.
If you want a good view of the Statue of Liberty and the harbor for free, ride round-trip on the Staten Island Ferry.
http://www.nyc.gov/html/dot/html/ferrybus/statfery.shtml#...
Free theatre: http://www.publictheater.org/view.php?mode=eventdisplay&a...
It was always fun on a Saturday morning to grab the morning paper and some coffee and go sit on the sidewalk and wait in line for tickets. Will also be a great way to chat w/some New Yorkers and get advice on what to do and where to go.
Central Park - get yourself a picnic from a deli and spend the afternoon. Also a great way to chat w/New Yorkers and get advice to what to do and where to go.
If you really want to keep it cheap, I'd suggest picking up a copy of "Let's Go New York City" or "Lonely Planet New York City" (backpackers' guides.) They'll have tons of up-to-date suggestions (and reviews/ratings) of out-of-the way, unexpected places - and you can then look for stuff that's more tailored to your interests.
Also, as you get closer, check out websites for Central Park, Bryant Park, and Prospect Park (in Brooklyn.) There's tons of free stuff that goes on in the parks in the summer.
When you get to the hotel, ask your concierge about what street festivals are going on - that's a great way to see a new neighborhood, do some deal-shopping, eat some good food, people watch. And there's always some festival SOMEWHERE during the summer.
General advice on being in NYC: People are ridiculously friendly, more than happy to help you find places or offer advice. New Yorkers love New York and want you to love it, too.
TAKE THE SUBWAY!!!!! Unless you're in a wicked hurry AND there's no traffic (never) - or unless it's 3 am - don't take a cab, TAKE THE SUBWAY. It's safe, it's easy, it's cheap, you can get ANYWHERE you need to go. If you think you'll be taking it a lot, you can buy a weekly pass - last I was there, if you commuted to work (so, 10 rides/week) it paid for itself. So if you think you'll be taking the subway more than ten times (each of you, you share a weekly pass), it's worth it. (If you buy a 10-ride or 20-ride card, you can share those.) I mean, be responsible about your belongings; if it's later in the evening, get on a car with more rather than fewer people (if you're feeling nervous, get on the car the conductor is in, usually right in the middle of the train.) But there's no reason to be afraid of it.
New York is a wild and wonderful, warm and friendly city - especially in June! Weather will be lovely, you're going to have a great time. Best of luck!