First of all, congrats on getting your trip to NY squared away - easily the greatest city of them all, bar none.
I'm a seasoned NY vet, heading over there usually around every 12 months, though TBH it's been a bit more than that recently as we decided to go to Vegas last year. However we will be looking at heading to NY again in September time. We usually head for at least 5 nights as it gives us a decent amount of time to see things and have a bit of breathing space, to chill out, do some shopping, explore different neighbourhoods, etc. However, I always recommend to friends who head over there for the first time that 3 nights is ample - like any major city, it's easy to get overwhelmed quickly so 3 nights gives you the opportunity to dip in to NY, see the major sights and get out before it starts becoming a bit much.
A lot of what to do there can depend on things like the time of year you are going (it is properly freezing in winter/early spring and oppressively hot in midsummer), how long you are there for, your budget, if you're there with the wife or are the kids going too and what you want to see and do. Bear in mind that the big attractions are obviously very popular so be prepared to spend a decent chunk of your 3 days there queuing. For a 3 nighter, the attractions I recommend are:
- Empire State Building (though, IMHO, you get a better view at the Rockerfeller Centre with a smaller queue).
- Central Park (fantastic zoo there at W 72 Street).
- Natural History Museum (free to get in, however they do have a suggested donation).
- The Metropolitan Museum of Art (The Met)
- Statue of Liberty and Ellis Island (can be done on the same trip).
- Brooklyn Bridge.
- 9/11 Museum.
- Grand Central Station.
Think of Manhattan as a collection of villages, each with it's own atmosphere and vibe. My personal favourites are Little Italy and Greenwich Village, which is very green and has a vast selection of restaurants and bars that are quite chilled out and along with the East and West Villages, offer something a bit different from central Manhattan. If you want a great club scene and some top bars, the Meatpacking District is the place to go, though Tribeca is good too. As Stevienics pointed out, Harlem is a great area which is clean and safe and has a terrific vibe to it. If you have time, I recommend a trip to Brooklyn as well. Brooklyn is the NY equivalent of being 'South of The River' in London and is probably closer to your idea of what New York actually is. It's an up and coming area with plenty to see and do and dead easy to get to.
As for getting around, the subway map isn't as easy to negotiate as the London Underground, though Manhattan has plenty of trains and lines to get you all over the island. Taxis are plentiful and cheap. You can get a 5 or 7 day pass for the subway, though I can't remember how much this costs. However it is great value. The hop on and off bus tours are very good too for getting your bearings.
As for food, in NY you can get anything at anytime - if you want a Kazhakstani takeaway at 0400 Xmas morning, you can get one. NY is famous for its breakfasts and there are a plethora of diners all over the city to get a decent brekkie that's reasonably priced. Likewise for lunch, we tend to go to a deli and stock up on sandwiches, fruit, juice and water to see us through.
As for military discounts, I will admit I have not had much success with this as discounts are there for US military. I've still to get one yet for the Empire State Building, though I've managed to get a few times for the Statue of Liberty, however I think it was down to long queues behind me and the people on the desk not wanting any hassle so they have given them to me. I know of others who have managed discounts at other places so I think I've just been unlucky - like every where else, if you don't ask, you don't get.
There is an abundance of entertainment in the evenings, with theatre, comedy, concerts galore. On Times Square, there's a TKTS stall which does discounted tickets for Broadway shows (if you aren't bothered about what you want to see it is great value, otherwise it's probably best to book ahead if you want to see a particular show).
To be honest, I could go on all day about NY, I love it so much. You've deffo fallen on your feet bagging a trip there and hopefully it will give you the impetus to go back again. We fell in love with it almost straight away and is, as I said, hands down the best city break of them all. If you need any more info, PM me.