Skip the Statue of Liberty, since you can't even go up in it now. Instead of the Empire State Building, try going to the top of Rockefeller Center (Top of the Rock). As someone else mentioned, the Bodies exhibit is here in the summer. I'd stay away from Chinatown unless you need to get some knockoff purses, belts, or watches as gifts.
Many of the museums have free admission on Fridays, although they can be very packed and close early. If you want to venture way uptown, you can visit the Cloisters, which is part of the Metropolitan Museum of Art. If you pay there you can also get into the Met. They suggest $20 to get in but you can pay whatever you want, even $1. The park there is very beautiful and more quiet than Central Park.
Here's some places I love to eat:
Highline (Thai & killer Mojitos)
835 Washington St
New York, NY 10014
Meatpacking District at Little W 12th St
Rice to Riches (Rice pudding - freaking amazing!)
37 Spring Street (Btwn Mott & Mulberry)
New York, NY 10012
Neptune (Polish food - excellent & cheap)
194 1st Ave, New York 10009
At 12th St
Yakitori Taisho (very authentic food - place is a dive though - but in a good way)
5 St. Marks Pl.,
New York, NY 10003
between Second and Third Aves.
The Eatery (great for Sunday brunch. Gay crowd)
798 9th Ave
New York, NY 10019
I don't go out to the gay bars a whole lot anymore. The scene is pretty tired. Chelsea is very boring to me and Hell's Kitchen is getting that way as well. Avoid any place where you have to pay a large cover charge since most places are free. If you want to go dancing, Splash charges I think $20 but The Ritz is free.
I prefer going to the East Village - more down-to-earth people. A couple of guys have mentioned Phoenix, which is my favorite bar in the city. You can also go to Williamsburg in Brooklyn and check out Metropolitan. Many people like it. I like Therapy in Hell's Kitchen, mostly b/c my roommate works there and I know the staff. They have some fun shows on the weeknights.
Check out Beauty Bar close to Union Square. You can get a manicure and cocktail for $10 and it also has dancing.
If you have time, go to DUMBO in Brooklyn. It has lots of cute stores, restaurants, and coffee shops.
My favorite bookstore in the city is The Strand in Union Square (which is a great place to visit).
There are also many free things going on in the city. Check out Club Free Time at http://www.clubfreetime.com/new_york.asp
Also check out Time Out New York at http://www.timeout.com/newyork/
. It lists many great things to do.
In my opinion, Broadway is in a very sad state and you can save your money by skipping a musical based on a movie. There's not a whole lot of originality out there right now.
My other tip is to pick the one thing you absolutely have to do on a specific day and not try to pack everything in; the rest should be optional for the day. Everything takes much longer than you think it will take.
If you go out to the bars, take a specific amount of money and leave your credit card or debit card at home. Otherwise you'll come back to the hotel and wonder how you spent several hundred dollars. However, many bars have a happy hour that goes till 9 or 10 p.m. so you can get half-price drinks then.
Hope you have a great time!