New York is actually pretty easy to get around, because it's like a grid. Get an easy-to-read map and use GPS on your phone, and you'll be able to get around anywhere. Also, contrary to popular opinion, New Yorkers are pretty friendly, so don't be afraid to ask for directions.
Don't worry about getting in at 10PM. There's a line of taxis outside, so just join the others on line and you'll get one--you won't have to hail one down.
In general, taxis are easier, but they also get expensive. The subway can get a little confusing (the subway map they post in the stations is kind of hard to figure out)...but if you know exactly where you are going, you can ask your friend which subway line you should take (or even look it up online), and you'll be fine taking it (again, ask for directions if you need to!)
The subway is not dangerous (though it's not somewhere you want to be if it's 3 in the morning and you're alone). Also, just so you know, you can now pay by credit card in cabs (and if they say their machine is broken, they're lying, so just swipe your card anyway...though it's illegal for them to do that, so they don't really do it so much anymore!)
You'll be able to fill your time REALLY easily--and NY is great because you can kind of split your activities up so that you're not racing all over town--for example, there are a bunch of museums in walking distance of each other, so you can do that for a day instead of going to one museum uptown and then running to another downtown, and then back uptown again, etc.
Here are some things to do:
--"Museum Mile"--a mile-long stretch on the Upper East Side with some of the best museums in New York: the Guggenheim and the Met. If you walk over to Madison, you get hit the Whitney too. Central Park is also right there near Museum Mile, as is the entrance to the Central Park Zoo (totally worth seeing, trust me), so that's a whole day of activities within only a mile.
--Midtown. I personally hate Times Square because of the crush of people, but you kind of need to see it if you come to New York. Times Square has a bunch of its own attractions: Madame Tussaud's Wax Museum, Forever 21 (4 floors!). Also, in the nearby area, you can see Grand Central, the Plaza Hotel, Rockefeller Center (and go to the 'top of the Rock' if you want...I think the view is even better than the Empire State Building, which is also way more annoying/crowded), St. Patrick's Cathedral (a must! Right next to Saks Fifth Avenue), the Museum of Modern Art and the NY Public Library. That area also has lots of shops. Just walk up and down 5th Avenue and Madison in the 40s and 50s, and you'll hit tons of stores: Urban Outfitters, Saks Fifth Avenue, Joe Fresh, Zara, H&M, Tiffany, Bergdorf Goodman, the Apple Store, Sephora, Cartier, Prada, Barney's New York, Armani, etc. Honestly just TONS of shopping in that area.
--Columbus Circle. Again, fun shopping.
--Natural History Museum. On the Upper West Side. One of my absolute favorites
--There are lots of cheap, fun comedy shows. Caroline's on Broadway consistently has great comedians, as does the Upright Citizens Brigade (pretty famous place)
--Walk along the High Line. Info is here: http://www.thehighline.org/ and http://www.thehighline.org/about/maps
--See a Broadway show! Matinees are often cheaper, and you can go to the TKTS booth in Times Square to get cheap tickets (or you can just buy online too). Some of my favorites and/or more popular shows: The Lion King, Wicked, Jersey Bots, Chicago, Newsies, Once. Go to Broadway.com to see the shows.
--Go to the Intrepid Sea Air Space Museum. Really cool.
--Go to Lincoln Center, just because it's so beautiful to see. You could possibly catch a ballet. When you're there, go to P.J. Clarke's--a very popular place for (very good) burgers.
--Wander around SoHo. It's a pretty cool place.
New York is great for just wandering. You can walk in a random area and will inevitably find a museum, shop, restaurant or cafe that catches your eye.
Some random tips:
--Ellis Island and the Statue of Liberty are amazing, but as far as I know, they are both still closed because of Hurricane Sandy
--Whether on the East Side or West Side, don't go too far uptown. It's usually a good rule to stay south of 100th.
--DO NOT take pedi cabs (bicycle cabs). They charge random prices--and often raise the price you've agreed upon once you get to your destination. They are awful.