In New York, eat early or eat late. If you go around 7-9pm to ANY restaurant you're flat out fucked. Prepare to wait.
In Bushwick (an industrial part of Brooklyn that has an influx of artists and restaurants pushed out of the increasingly expensive Williamsburg and Greenpoint area) there are several great places springing up if you don't mind venturing outside of the tourist areas.
Rando food places I like:
Cafe Ghia - The best burger you will have in New York. I don't care what other people say. They are objectively wrong because they have most likely not eaten here. This is a beautiful tasting burger with great quality meat and amazing texture which is then injected with bacon fat previous to cooking making it taste like God just came in your mouth. On top of this you add slow braised bacon which is a quarter inch thick and you have the makings of a food experience you will most likely not top in New York for the cheap price. There is other food here and it is excellent as well.
http://www.cafeghia.com/The Rookery - Also in Bushwick, this is a combination Scottish / Caribbean food place with cheap drinks. I've only been disappointed by food here once when the weekly Sunday Roast was ehhhh. The Oxtail Sloppy Joe and Curried Goat Shepards Pie are highly recommended. The Mum's chips are to die for. Don't come here for brunch.
http://therookerybar.com/Roebling Tea Room - There is no bad food here. Period. If you come for bunch, they make the most amazing egg dishes. Unfortunately everyone knows this so it can get kind of packed but luckily they open fairly early. If you get there before noon, it should be fine. It's a good way to start a day in Brooklyn
http://www.roeblingtearoom.com/ (if this place is packed and you don't want to wait)
Miss Lilly's - If you find yourself in SoHo for some shopping, head to this Jamaican place. If it's lunch time, go around the corner to the Juice Bar / Cafe section where they have smaller portions / prices. I highly recommend the Jerk Chicken Salad while my GF swears by the Roti. This place has the hottest people I have ever seen working in a restaurant
http://www.misslilysnyc.com/assets/pdf/menus/misslilys-cafe-menu.pdfMighty Quinns - If you want BBQ, go here. There are other places in New York, but they're way over priced for what you get. Get the brisket and ask for it "moist" so you get the fattier cuts. It will literally melt in your mouth.
http://mightyquinnsbbq.com/Momofuku - There are a few varieties of this place. Momofuku Milk Bar is a desert place which has cookies and ice cream and shakes and stuff like that. People love it. Too much sugar for me, but vacation calories don't count. Momofuku Ssam Bar - come here for the duck lunch. Get one service to split between two people. This place is stupid delicious.
http://momofuku.com/new-york/ssam-bar/ http://milkbarstore.com/Chop't - This is a salad chain with locations all over midtown and the financial district. Very good very inexpensive (for NYC anyway) salads. If you're at the 9/11 memorial (good god, why?) or wall st (again, why? there's nothing there except herds of disappointed tourists) this is a good place for lunch. The lines can get long, but they move at a good clip. I eat here at least once a week.
http://choptsalad.com/Odd Fellows Ice Cream. do you like ice cream? go here. it will still be warm enough in early September to justify it.
http://www.oddfellowsnyc.com/flavors.htmlWill add more as I think of them.
I'm kind of at a loss for off the beaten path touristy stuff to do. I can tell you more what to avoid (Times Square is a large loud outdoor mall which is perpetually crowded). Highline is good, go at dusk for a beautiful walk. Start south, work your way north, get dinner at Chelsea Food Market
http://www.chelseamarket.com/Brooklyn Museum is a great place to visit and inexpensive and rarely crowded. This place tends to have actually interesting modern exhibits rather than focusing on post-war showcase exhibits like MoMA. Next door is the Brooklyn Botanical garden which is just wonderful.
Walking around central park is actually fun. It's huge and it's varied. I enjoy the Bronx Zoo. There is an add campaign for the Prospect Park zoo, and I have to say skip it. Coney Island Aquarium is sad and pricey (much like Coney Island which is losing the weird charm and being turned into a mall). New York Historical Society is very interesting as is the Brooklyn Reliquary Museum (filled with post war NYC artifacts packed like a crazy uncle's apartment)
I don't really know what you're looking for, so it's hard to suggest.
