NYC at Christmas is always an interesting place, the list of things to do is practically limitless.
Going in and out of New Jersey, albeit local can be a bit torturous especially during the Winter... Things will be more expensive during holiday time. I googled some stuff for you--
Inexpensive hotels (good if you're with more than yourself, and many of these hotels aren't bad)
And, some
Hostels (but I cannot vouch for any of these)
The only thing I can advise other than that are comfortable walking shoes, and if its rainy/slushy, waterproof comfy boots. Trust me... Wet Cold feet stink.
As for safety, the subway is rather safe at night, as is walking around most places. (Obviously, walking into a dark alley at 2am by yourself is a bad idea. Even if you were in podunk nowhere.
) Most locals are friendly, and will give you directions if you need.
At Christmas, you MUST go to Herald Square (its on 34th st.) and see all the storefronts, then, you MUST go to Rockefeller Center (5th Ave and 48th street, I think) to see the tree and go ice skating.
Broadway is the touristy thing to do, there are any number of shows and restaurants to go and see... The Tkts stand in Times square (47th and Broadway) is a popular place to pick up discounted tickets for shows... BUT... there's another location at the South Street Seaport (Corner of Front St. and John St.). It tends to be less crowded, but it works on a different schedule. If I'm not mistaken, they take cash and travelers checks only (BUT, someone told me they take credit cards)
The Statue of Liberty and Ellis Island are two very interesting things to see-- You have to make reservations to climb to the top of the statue, but if you have the stamina, its well worth it. Ellis Island is an excellent museum of American immigration.
The Museum of the American Indian is an offshoot of the Smithsonian, It's within walking distance of the Statue of Liberty Ferry, and the Staten Island Ferry(which to me is a means of commute, but its free and many people like to go and take it to go and sight see). This museum is very interesting, I love educational things.
Trapeze School is supposed to be lots of fun (you'd never catch me doing it. EVER)
My all time favorite Museum is the American Museum of Natural History (did I mention I am a nerd?), including a trip to the Hayden Planetarium at the Rose Center of Earth and space (the visual you get walking up 81st street to the museum during the winter looks like a gigantic ice cube just sitting outside of Central park. So beautiful.) On Friday and Saturday nights at I think 7:30, they have a program called SonicVision... Its about an hour long and is nothing but fantastic.
If you're into clubbing, depending on what you're into, there is somewhere to go and something to do every night until the wee hours of the morning (or the late hours of the morning
) Let google be your director. The only thing I warn is that they can be VERY expensive.
Even though You are from 'Big' Italy, you might enjoy Little Italy, its definitely a very American place... its bordered by Chinatown, which is another great place to just walk around during the day and night.
That's all I can think of at the moment... Hope it helps!
I hope you enjoy your vacation!