Fun fact: You can get into the Met and the Museum of Natural History for free. There's a suggested donation of like 10 bucks, but you can donate as much or as little as you like. While there are other museums in New York worth your time -- MoMa, the Guggenheim, and I'm a big fan of the Frick -- but those two you can waste literally days at for nothin'.
Depending on what the weather's like, I would say the best thing you can do is just walk around, see what you find. Get one of those Streetwise maps that's laminated and fits in your back pocket. Don't be afraid to take the subway. Remember that the more you get down into the Village and Chinatown, the less organized the streets are. Embrace the Grid. Pick a street -- Broadway and Fifth Avenue are good bets -- and just walk up or down it. See what you find. Be on the look out for plaques telling you "something happened here!" That's how I found the house where Herman Mellville wrote Billy Budd and the Triangle Fire building. Walk across the Brooklyn Bridge at or just after sunset.
If you're interested in seeing a Broadway show, TKTS is the way to go, but there's one at the South Street Seaport that doesn't have the lines of the one in Times Square. Also, keep in mind that it'll be easier to see a play than it will be to see a musical. If there's a specific show you want to see (I highly recommend next to normal, but only if you have a couple of hours to recover afterwards), see what that show's rush tickets policy is. Generally you can get cheaper tickets that way.
You're a crime fiction guy, so Partners & Crime on Greenwich Street is a must. They're really nice and if you prove your stuff, they might give you a free book. The Strand on 12th and Broadway is the one of the largest used bookstores in the world, although I'm not crazy about their mystery selection, but that's a must-do. Kim's Video on 1st has moved around a bunch and closed some of their most famous branches, but you might find some good deals on obscure films. I love, love, love Shakespeare and Co. on Broadway, too.
The Zigfield's a great theater and a New York landmark, but I'd also recommend checking out what's playing at Film Forum that weekend. The theaters aren't stadium seating, but there's always something cool going on there, and it's one of the last major rep houses in the city. If you're there on the weekend, see what's going on at the IFC Center, too -- they generally have some awesome midnight movies and matinees, as does the Sunshine. The Angelika's the closest thing we have in the city to the Alamo Drafthouse (and it's pretty far from it), and you kind of have to live with the fact that it's right by the subway line. Eat at Two Boots Pizza, for my money, some of the best pizza in New York. Their "The Dude" slice is to die for.
This was kind of Village-centric, but I might drop in with some more ideas later. Do you want to stay at a hotel, or would you be up for staying at a hostel-type place? I've stayed at a number of those and could give you some recommendations. I'll leave you with this:
If the street is clear and there are no cars coming, it's safe to walk across the street. Don't be one of those tourists waiting for the light to change. Follow the crowd, and remember, a car may hit one person, but they're certainly not going to hit two people.
